Lower Bound methods for the Shakedown problem of WC-Co composites

Lower Bound methods for the Shakedown
problem of WC-Co composites
Basavaraju Akula
August 2014
Mastert Thesis
Institut für Werkstoffanwendungen im Maschinenbau
RWTH Aachen University
Supervisor 1: M.Sc. Geng Chen

Contents
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Incremental Methods and Direct Methods . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Overview of current Research state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Thesis scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Micro Mechanics 5
2.1 Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Particulate reinforced composite materials . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Micro-mechanical Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Material Randomness and RVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 Effective properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6 Cemented Carbides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.6.1 Classiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.6.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Optimization 13
3.1 Convex and Non-Convex Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.1 Convex and Non-Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 Optimization Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3.1 Penalty Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3.2 Augmented Lagrangian method . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3.3 Interior point method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4 IPOPT(Interior Point OPTimizer) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
i

CONTENTS ii
3.4.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4.2 Interpreting the output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4 Shakedown Theory 22
4.1 Material Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 Shakedown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3 Kinematic Shakedown theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.4 Static Shakedown theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.5 Proof of theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.6 Von mises yield criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.7 Loading Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5 Damage 32
5.1 Damage representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.2 Effective Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.3 Constitutive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.4 Damage parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.5 Damage curve ﬁtting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6 MATLAB implementation 39
6.1 MATLAB Interface for IPOPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6.1.1 IPOPT function call . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6.1.2 funcs MATLAB struct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.1.3 options MATLAB struct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.2 Symbolic Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
7 Validation of the developed numerical method 42
7.1 Square plate with a hole under a biaxial load system . . . . . . . . . . . . . . . . . . 42
7.2 Elastic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
7.2.1 Pressure Px=100 MPa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
7.2.2 Pressure Py =100 MPa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7.2.3 Pressure in x,y directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
7.3 Limit load with Incremental Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7.4 Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

CONTENTS iii
7.5 Considering the Damage of material . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7.6 Residual stress ﬁeld visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
8 Case Studies 52
8.1 Image Recognition and mesh generation . . . . . . . . . . . . . . . . . . . . . . . . . 52
8.1.1 Image Toolbox: MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
8.1.2 Image clean-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
8.1.3 FE mesh technique: ABAQUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
8.2 RVE Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
9 Descriptive statistics 57
9.1 Correlation plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
10 Conclusion 61
A MATLAB Symbolic toolbox 62
B IPOPT Code 66

List of Figures
2.1 SEM image of a sample RVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Material randomness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Effective properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1 Convex Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Non convex Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Maximum and Minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.1 Elastic Shakedown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Alternating Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3 Ratcheting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.4 Von mises yield criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.5 Loading polyhedron in a three dimensional loading space . . . . . . . . . . . . . . . 29
5.1 Physical damage and mathematical continuous damage [25] . . . . . . . . . . . . . 33
5.2 Damage function ﬁtting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
7.1 Square plate with a hole under a biaxial load system . . . . . . . . . . . . . . . . . . 43
7.2 Finite element discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
7.3 Pressure Px=100 in x-direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
7.4 von Mises stress with Px=100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7.5 Pressure Py =100 in y-direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7.6 von Mises stress with Py =100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
7.7 Pressure in x- and y- direction with Py =Py =100 MPa . . . . . . . . . . . . . . . . . . 46
7.8 von Mises stress with Py =Py =100 MPa . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7.9 Elastic Stress superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
iv

LIST OF FIGURES v
7.10 Graphical plot for the various Limit and Shakedown load factors . . . . . . . . . . . 49
7.11 Visualization as a temperature field in ABAQUS (Limit Analysis) . . . . . . . . . . . 50
7.12 Visualization as a temperature field in ABAQUS (Shakedown) . . . . . . . . . . . . . 51
7.13 Visualization as a temperature field in ABAQUS (Damage) . . . . . . . . . . . . . . . 51
8.1 Overview of the automated technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
8.2 RVE mesh in ABAQUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8.3 RVE mesh in ABAQUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
9.1 40µmx40µm RVE data( ΣU
,µ, E, WC Vol %) . . . . . . . . . . . . . . . . . . . . . . . 58
9.2 40µmx40µm RVE data( Σ∞
,µ, E, WC Vol %) . . . . . . . . . . . . . . . . . . . . . . . 60

LIST OF FIGURES vi
Abstract
Most mechanical components are subjected to cyclic loadings during their life time. Predict-
ing the life of such structures and components is of vital importance. The safety margin of a
structure against accumulated inelastic deformations play a critical role in design of such com-
ponents. Direct method(DM) based on the bounding theorems proposed by Melans and Koiter
provide means towards this end. According to DM, the original structural problem can be for-
mulated numerically into a nonlinear programming whose solution yields the structure’s shake-
down limit.
In present thesis, to predict the strength of Particulate Reinforced Metal Matrix Composites
(PRMMC), the shakedown problem is modeled using MATLAB and solved by a nonlinear op-
timization solver Ipopt (Interior Point OPTimizer). To account for the phenomenon of material
degradation, constitutive law of ductile damage has been integrated with the DM. Using such
approach, the strength of few PRMMC samples are determined and interpreted statistically. Few
performance indicators have been identiﬁed from results, applying these indicators we investi-
gated the underlying factors that inﬂuence the strength of material.

Chapter 1
Introduction
1.1 Background
In the broadest sense every material is expected to fail when loaded beyond a certain limit. The
classical elastic theory states that when the stress intensity attains the yield point stress, the
material would fail. Experimental evidences conﬁrm that even after the stress intensity reached
the yield threshold, it can deform considerably without breaking.
This implies that the structure does not necessarily fail or deform excessively. Instead, a certain
amount of stress redistribution takes place and further load increments can be supported. This
redistribution of stresses caused by plastic deformation occurring in loading cycles may be such
that in subsequent loading, the members of the structure may be operating solely in the elastic
domain for higher load levels. This is the basic idea of shakedown.
1.2 Incremental Methods and Direct Methods
Solution to most of the engineering problems is based on linear approximations. Below are the
approximations.
• displacements are small and can be neglected in equilibrium equations
• the strain is proportional to the stress (linear Hookean material model)
• loads are conservative, deterministic and independent on displacements
• supports of the structure remain unchanged during loading
1

CHAPTER 1. INTRODUCTION 2
For non-linear static analysis, the loads are applied in incremental steps using time curves. The
“time” variable represents a pseudo time, which denotes the intensity of the applied loads at
certain step. A typical incremental FE method deals in mechanical problems with short and
deterministic loading. Long and stochastic loadings are confronted with computational diffi-
culties, and cannot be modeled within the framework of the incremental methods. This hinders
the use of shakedown limit as design criteria for structural engineering problems. DM’s provide
a solution to this problem.
DM’s can be conceptually regarded as a generalization of the classical limit, rigid-plastic analy-
sis. Contrary to the evolutive elasto-plastic analysis, it does not necessarily require full informa-
tion on the history of loads, thus allowing direct evaluation of the structure’s ability to adapt to
given bounded loading envelope.Unlike the conventional elastic analysis in which only the most
critical combination of working loads are used to asses the structural safety, the DM is a more
intuitive approach looking at a bounded loading envelope, whose loads can possibly influence
the safety factor.
The idea of shakedown analysis goes back to the 1920’s and was first formulated in its general
form by Melan [28] and reformulated later by Koiter [20]. Melan proposed a theorem based
on a series of static conditions predicting the lower bound of a structure’s loading capacity. By
implementing this theorem into the FE method, the problem of finding the shakedown limit of
an actual structure can be formulated into a corresponding large-scale nonlinear programming
problem, whose solution yields the structure’s shakedown limit under cyclic loading [37].
1.3 Overview of current Research state
The plastic strains play an important role in determining the strength of material. The materials
ability to confine these deformations significantly determines its usage. Composite materials,
similar to the single phase material, can be characterized by three strength gauge parameters:
yield strength, ultimate strength and endurance limit. They correspond to elastic limit, plastic
limit and shakedown limit of the material, respectively.
With rapid advancements in the filed of computation power available at hand, the recent
decades have seen significant progress in the field of DM. A number of studies have been made
in order to implement the DM’s using the bounds, as stated by Melan-Koiter [28] [20] to evaluate
the strength of heterogeneous material. These bounds can be broadly classified into two cate-

gories namely, the Lower bound also called as the static bound and the upper bound also called
the kinematic bound.
Studies related to the prediction of Lower bounds for the materials have been of paramount
significance. Among them, we would like to name Weichert and Schwabe who contributed to-
wards shakedown analysis of composites with periodic micro-structure [36]. Magoariec [4] de-
veloped direct numerical approach for elastic plastic shakedown of 3D periodic heterogeneous
media.Zhang, H. and Liu, Y. and Xu, B [39] implemented plastic limit analysis of ductile com-
posite structures.J.-H. You [38] worked on Shakedown analysis of fiber-reinforced copper matrix
composites by direct and incremental approaches. M. Chen [9] contributed towards the shake-
down analysis of periodic composites.
And in contrast, to the Lower bound approaches contributions have been also made towards the
upper bounds. Carvelli [6] implemented upper bound approach to the shakedown analysis of
unidirectional fiber reinforced metal matrix composites. H.F. Chen and Ponter [8] studied on the
behavior of a particulate metal matrix composite subjected to cyclic temperature and constant
stress, H.X. Li [27] developed a microscopic nonlinear programming approach to shakedown
analysis of cohesive-frictional composites, and Barrera [1] dealt with the application of linear
matching methods to do shakedown analysis of heterogeneous materials.
Efforts have been made to extend the shakedown theory to damaging materials, materials that
deteriorate in terms of stiffness and resistance during the straining process. Contributions by
Polizzotto [5] towards the elastic-plastic-damage materials, in the presence of a material model
exhibiting a multiple dissipative mechanism (plastic strain, damage, friction) the classical no-
tion of shakedown needs being suitably widened.
For the class of elastic–plastic materials (metals and alloys), which exhibit ductile damage, the
damage mechanism is driven by the plastic strains [24]. Once the plastic strains cease the dam-
age is no longer a part of the deformation process. In such cases the basic shakedown notion
can be extended to an elastic–plastic material with damage induced by plastic strains.
Significant progress has been achieved in this respect by Hachemi and Weichert [12] in which
the classical static shakedown theorem is applied to a generalized standard material that ex-
hibits ductile isotropic damage according to a damage model of Lemaitre and Chaboche [24].
This work is further elaborated by Hachemi and Weichert [13] and Druyanov and Roman [11].

1.4 Thesis scope
The DM’s involve formulation of the shakedown problem into a corresponding large-scale non-
linear programming problem. The regular mathematical programming method’s face difficul-
ties when model size grows. This hinders the existing studies from dealing with the complicated
micro structures and retort to rather idealized micro structures. Towards this end one has to rely
on more able and competent optimization algorithms.
Interior-point method (IPM) introduced by Karmarkar [19] during 1980’s demonstrated incred-
ible efficiency in solving linear programming problems. Unlike the Simplex methods, iterates
are calculated not on the boundary, but in the interior of the feasible region. Additionally the
iteration complexity and count is much better than that of conventional algorithms. Most of
the modern optimization software packages contains IPM version of Linear programming (LP)
and many of them have IPM codes for various non-linear problems such as convex quadratic,
semi-definite, and cone programming, to mention just a few.
In current study a WC- 70 Wt. % Co PRMMC is considered for load capacity against both long
and short-termed loads. Modelling the micro structure using the technique developed by G.
Chen [7] helps in dealing with realistic micro structures. Using this technique a series scanning
electron microscope (SEM) images derived from the material are converted to finite element
(FE) models. The static limit and shakedown problems are constructed on each representative
volume element(RVE) and solved by Ipopt a interior-point method based solver.
The damage constitutive model by Hachemi and Weichert [12] is included into the shakedown
problem formulation. The effect of the damage variable on the load factor is addressed consid-
ering various RVE’s from the composite.

Chapter 2
Micro Mechanics
2.1 Composites
In the construction of heterogeneous materials, the basic philosophy is to select material com-
binations to produce desired aggregate responses. This leads to inhomogeneities.Composite
material consists of two or more materials fabricated in such a way that individual constituents
are readily identified in the final form. If designed properly, this combination will retain the
optimal properties of each constituent, or even some advantageous properties not exhibited by
the individual constituents. The composites generally consist of two phases.
• Matrix phase The primary phase, having a continuous character, is called matrix. Matrix
is usually more ductile and less hard phase. It holds the dispersed phase and shares a load
with it.
• Dispersed phase The second phase is embedded in the matrix in a discontinuous form.
This secondary phase is called dispersed phase. Dispersed phase is usually stronger than
the matrix, therefore it is sometimes called reinforcing phase.
Composites can be classified based on the Matrix material and the reinforcing material struc-
ture.
Based on Matrix material
1. Metal Matrix Composites(MMC)
2. Ceramic Matrix Composites (CMC)
5

CHAPTER 2. MICRO MECHANICS 6
3. Polymer Matrix Composites (PMC)
Based on Reinforcing material structure
1. Particulate Composites
2. Fibrous Composites
3. Laminate Composites
Assessment of the macroscopic properties of composites is one of the major concerns in mate-
rial and structural engineering. As far as the elastic properties of the composites are concerned,
the micro mechanical techniques are nowadays widely applied and proved successful. Never-
theless, the application of the micro mechanical tools to the modeling of the non-linear behav-
ior of composites is relatively recent and several issues are still open. In particular, the determi-
nation of the strength properties of randomly heterogeneous materials remains an important
task.
The recent years has seen a exponential increase in the computational power available for math-
ematical modeling and simulation. This raises the possibility that modern numerical methods
can play a significant role in the analysis of heterogeneous micro structures.
2.2 Particulate reinforced composite materials
Particulate Composites consist of a matrix reinforced by a dispersed phase in form of parti-
cles. Advanced particulate reinforced composite materials have been widely used in various
load bearing structures, from sporting goods to aerospace vehicles. The increasing popularity
of these composites are largely due to their light weight, high strength and superior structural
durability. Besides, many other functional properties can be tailored through a proper com-
position of the inclusion and the matrix. The spatial layout of the inclusions in matrix con-
solidated composites often forms a complex network, an integral part of the composite micro-
structure. This micro structure plays an important role in the composite properties.
PRMMCs generally appreciate higher isotropy and demand less effort to be manufactured
[17].The randomness of PRMMC’s pose a significant challenge in guaging the global response
of the composites. A statistical analysis of RVE models which are samples of the micro structure
large enough to suitably reflect the stochastic fluctuations of material properties on a pertinent

scale can be a possible solution. The randomness of micro structure also introduces difficulties
in the numerical representation of the material. The modeling technique of the micro structural
geometry and the size effect of RVE, have been extensively investigated in many studies [3] [10].
Figure 2.1: SEM image of a sample RVE
2.3 Micro-mechanical Homogenization
One important goal of the study on the heterogeneous materials is to derive their effective prop-
erties. Generally these properties depend on the constitutive laws and spatial distribution of
their components. In principle, property characterization of composites should consider the
micro structures. Due to difficulty of modeling, the true micro structure is often replaced by the
simplified idealization. However, the simplification would entail inaccuracy of the predicted
properties.
There are numerous benefits of analytical methods, one is that they are able to deliver rigorous
bounds for the macroscopic linear properties. These bounds include the well known Voigt and
Reuss bounds which take only the volume fraction of the components into account; also Hashin
and Shtrikman’s [14] bounds incorporate the notion of isotropic distribution of phases. Further
bounds, in the case of random media, were obtained in the general case by Beran [2], and later
for two-phase materials by Milton [29]. The incorporation of more and more statistical informa-
tion on the distribution of heterogeneities in random materials leads to a hierarchy of bounds, as
suggested by the systematic theory of Kröner [22] , and also in Torquato and Stell [34], Torquato
and Lado [33]. Some of these bounds are optimal in the sense that specific morphologies can be
designed that gives exactly the value of the bound as effective property.
In additional, analytical methods can provide direct estimations of the interested effective prop-

erties. In this respect, Mori-Tanaka model favors one phase as a matrix. In contrast, the self-
consistent (SC) scheme, presented by Beran [2] for thermal conductivity and by Hershey [15]
and Kröner [23] for linear polycrystals, refers to a disordered distribution of phases. A geometri-
cal construction given by Milton for two-phase composites is obtained by a multi-scale stacking
of spheres of every phase with the appropriate volume fractions. Here the role of every compo-
nent is considered symmetric for the estimations of SC model.
Within these theories, the proposed estimations are valid for random composite with an infi-
nite domain, therefore they and can be denoted as asymptotic estimates. Another way to treat
homogenization problems is to use numerical techniques and simulations on samples repre-
senting the micro structure. In this case, the definition of RVE is of paramount importance. The
prediction of the macroscopic stress-strain response of composite materials depends on their
micro mechanical behavior exemplified by the interaction between the constituents. In this
context, the micro structure of the considered material should as well be taken into account.
2.4 Material Randomness and RVE
During the modeling of RVEs, complications arise once the geometry of inclusions apparently
differs from idealized geometries. This difficulty can be resolved by analyzing RVE models which
are samples of the micro structure large enough to suitably reflect the stochastic fluctuations of
material properties on a pertinent scale. The restriction of computational feasibility requires
the RVE size to be as small as possible.
RVE is a sample (partial) volume of the composite that contains a sufficient number of inclu-
sions to describe the heterogeneity of the composite. Several such volume elements of the com-
posite are analyzed statistically for fine predictions of the mechanical properties. The concept
of RVE is introduced based on deterministic continuum theories, therefore it excludes the ran-
domness of the micro structures which are, in fact, prevalent in real materials. The statistical
treatments using several RVE’s can be carried out as a bridge between micro to macro levels.
Here comes the importance of selecting the right RVE size which is small enough and depicts
the heterogeneities at the micro level.
There are many existing criteria for the determination of the RVE size. Among them the Hill’s
[16] criterion provides an insight from the energy view point. Another important criterion for
the linear elastic composite is from Drugan and Wills [35], which determines the minimum size

by comparing the magnitude of the non-local to local terms in their newly defined constitutive
equation and claims that the minimum RVE size is at most twice the reinforcement diameter for
any reinforcement concentration level. Drugans criterion was approved in several lateral stud-
ies, but disproved among others which involved plasticity of the matrix and more randomized
distribution of the inclusions. Furthermore, it has been proven by Ostoja-Starzweski [30] that
a minimum RVE can be determined only when either composite can be embodied by unit cell
with periodic micro structure or the volume containing a very large set of microscopic elements.
Some researchers see the need to adopt governing parameters other than the elastic moduli. In
this context, Stroeven [32] proposed key figures such as peak load, dissipated energy and strain
concentration factor and among others also peak stress and experimental strain distribution. In
this sense, as concluded by Stroeven, the determination of the RVE size is by no means straight-
forward. It not only depends on the material under consideration, but also on the structural
sensitivity of the measured physical quantity. Therefore there does not exist a uniform criterion
determining a minimum RVE size, factors like morphology of the micro structure, material be-
havior and purpose of application have significant impact on the determination of RVE. This
objective oriented mechanism is strongly approved by Kanit et al. [18] based on their study
about a random composite.
(a) RVE1 (b) RVE2
(c) RVE3 (d) RVE4
Figure 2.2: Material randomness

2.5 Effective properties
With homogenization a body of a heterogeneous material is transformed into a constitutively
equivalent body called “effective medium” characterized by “effective properties”. The theory
links the properties in micro-scale y and the macro scale x.
Figure 2.3: Effective properties
The mechanical properties of micro heterogeneous materials are characterized by a spatially
variable elasticity tensor IE. In order to characterize the effective macroscopic response of such
materials, a relation between averages is sought.The macro-stresses and macro-strains, which
the characterize the mechanical state of the macroscopic material point, are defined as the vol-
umetric averages of the microscopic fields.
Σ =
1
Ω Ω
σ(y)dv (2.1)
Here Ω is the RVE domain. The macroscopic strain can be defined as.
E =
1
Ω Ω
ε(y)dv (2.2)
Σ and E are corelated by the effectiv elastic stress tensor C
Σ = C : E (2.3)

2.6 Cemented Carbides
Tungsten mono-carbide usually referred to as tungsten carbide was discovered by Henri
Moissan in 1893 during his search for a method to make synthetic diamonds. It was then ob-
served that the hardness of WC is comparable to that of diamond. This material, however,
proved to be so brittle that its commercial use was seriously limited. Subsequently, research
was focused on improving its toughness, and significant contributions to the development of
cemented carbides were made in the 1920’s by Karl Schröter. Using cobalt (Co) as a binding
material, Schröter developed a compacting and sintering process for cemented tungsten car-
bides (WC-Co) that is still widely used to manufacture WC-Co composites. Most of the further
developments were modifications of the Schröter’s process, involving replacement of part or all
of the WC with other carbides, such as titanium carbide (TiC), tantalum carbide (TaC), and/or
niobium carbide (NbC).
2.6.1 Classification
• WC-Co grades : Straight grades, sometimes referred to as unalloyed grades, are nomi-
nally pure WC-Co composites. They contain 3-13 w/o (weight percent) Co for cutting
tool grades and up to 30 w/o Co for wear resistant parts. The average carbide particle
size varies from sub-micron to eight microns. Straight grades are used for machining cast
iron, nonferrous alloys, and non-metallic materials, but generally not for the machining
of steel.
• Alloyed grades : Also referred to as steel cutting grades, or crater resistance grades, which
have been developed to prohibit cratering during the machining of steel. The basic com-
positions of alloyed grades are 3-12 w/o Co, 2-8 w/o TiC, 2-8 w/o TaC, and 1-5 w/o NbC.
The average carbide particle size of these grades is usually between 0.8 and 4 µm
2.6.2 Application
This material shares few advantages of WC, such as high stiffness and high wear resistance.
Meanwhile, the brittleness of WC is partially compensated by cobalts ductility, thereby resulting
in higher toughness and durability.
The cemented carbides are primarily used as cutting tools. The most wear resistant materials are

the straight WC-Co grades which are used for machining and cutting of materials that require
abrasion resistance; e.g. cast iron and nonferrous alloys. On the other hand, the alloyed grades
are used for strongly yielding materials that require high deformation resistance, toughness, and
thermal shock resistance. Also, many applications exist for cemented carbides in non-cutting
areas such as mining, construction, and wear resistance components

Chapter 3
Optimization
3.1 Convex and Non-Convex Optimization
A convex optimization problem is a problem where all of the constraints are convex functions,
and the objective is a convex function if minimizing, or a concave function if maximizing. Linear
functions are convex, so linear programming problems are convex problems. Conic optimiza-
tion problems ,the natural extension of linear programming problems are also convex problems.
In a convex optimization problem, the feasible region, the intersection of convex constraint
functions is a convex region.
With a convex objective and a convex feasible region, there can be only one optimal solution,
which is globally optimal. Several method, notably Interior Point methods will either ﬁnd the
globally optimal solution, or prove that there is no feasible solution to the problem. Convex
problems can be solved efﬁciently up to very large size.
A non-convex optimization problem is any problem where the objective or any of the con-
straints are non-convex.Such a problem may have multiple feasible regions and multiple locally
optimal points within each region. It can take time exponential in the number of variables and
constraints to determine that a non-convex problem is infeasible, that the objective function is
unbounded, or that an optimal solution is the "global optimum" across all feasible regions.
Because of their desirable properties, convex optimization problems can be solved with a va-
riety of methods. But Interior Point or Barrier methods are especially appropriate for convex
problems, because they treat linear, quadratic, conic, and smooth non-linear functions in es-
sentially the same way they create and use a smooth convex non-linear barrier function for the
13

CHAPTER 3. OPTIMIZATION 14
constraints, even for LP problems.
3.1.1 Convex and Non-Convex Functions
Geometrically, a function is convex if a line segment drawn from any point (x, f(x)) to another
point (y, f(y)) , called the chord from x to y lies on or above the graph of f.
A non-convex function "curves up and down" – it is neither convex nor concave. A familiar
example is the sine function
Figure 3.1: Convex Function
Figure 3.2: Non convex Function

3.2 Problem formulation
Optimization can be applied to existing or specifically constructed mathematical models. The
idea is that one would like to find an extreme of one output of the model by varying several pa-
rameters or variables. The usual reason to find appropriate parameter values is due to decision
support or design optimization.
The main terminology in optimization is as follows. Usually quantities describing the decisions
are given by a vector . The property (output) of the model that is optimized (costs, CO2 emission,
etc.) is put in a so-called objective function f(x). Other relevant output properties are indicated
by functions gi (x) and are put in constraints representing design restrictions such as material
stress, gi (x)≤ 0 or gi (x) = 0. The so called feasible area that is determined by the constraints is
often summarized by x ∈ X. In nonlinear optimization, or nonlinear programming (NLP), the
objective and/or constraint functions are nonlinear.
The general principle of NLP is that the values of the variables can be varied in a continuous
way within the feasible set. To find and to characterize the best plan, we should define what is
an optimum, i.e., maximum or minimum. We distinguish between a local and global optimum,
as illustrated in Figure3.3
Without loss of generality the general NLP problem can be written as
min f (x),
s.t. gi (x) = 0, i = 1,.....,p
gi (x) ≤ 0, j = p +1,.....,q
ak ≤xk ≤ bk
(3.1)
where x = [x1, . . . ,xn] is a vector of n variables, f (x) is the objective function, hi (x) (i =1, . . . , p)
is the ith
equality constraint, and gj (x) (j = p + 1,. . . ,q; q≤n) is the jth
inequality constraint. The
ak and bk denote the lower and upper bounds of the variable xk(k = 1, . . . ,n), respectively. It is
assumed that all problem functions f (x), hi (x), and gj (x) are twice continuously differentiable.
In most of the nonlinear programming problems f (x), h(x), and g(x) are non convex and the
problems have multiple locally optimal solutions. In the only case where the f (x) is convex ,
every hi (x) is linear and every gj (x)is convex, constrained local minimum is also constrained
global minimum

Figure 3.3: Maximum and Minimum
3.3 Optimization Strategies
Methods for solving a constrained optimization problem in n variables and m constraints can
be divided roughly into four categories that depend on the dimension of the space in which
the relevant algorithm works. Primal methods work in n – m space, penalty methods work in
n space, dual and cutting plane methods work in m space, and Lagrangian methods work in
n + m space. Each of these approaches are founded on different aspects of NLP theory. Each
of these algorithms have a strong interconnection between them, both in the ﬁnal form of im-
plementation and in performance. The rates of convergence of most practical algorithms are
determined by the Hessian structure of the Lagrangian, much like the Hessian structure of the
objective function determines the rates of convergence for most unconstrained methods.
3.3.1 Penalty Methods
Penalty methods belong to the ﬁrst attempts to solve constrained optimization problems satis-
factorily.The penalty method replaces a constrained optimization problem by a series of uncon-
strained problems whose solutions ideally converge to the solution of the original constrained
problem. The unconstrained problems are formed by adding a term, called a penalty function,
to the objective function that consists of a penalty parameter multiplied by a measure of viola-

tion of the constraints. The measure of violation is nonzero when the constraints are violated
and is zero in the region where constraints are not violated.
The resulting unconstrained nonlinear programming problems are solved by any standard tech-
nique, e.g. a quasi-Newton search direction combined with a line search. However, the line
search must be performed quite accurately due to the steep, narrow valleys created by the
penalty terms.
min f (x),
s.t. ci (x) ≤ 0 ∀i ∈ I
(3.2)
The constrained optimization problem in 3.2 is reformulated as
min Φk(x) = f (x)+σk min(0,ci (x))2
(3.3)
Here σk are the penalty coefﬁcients. In each iteration k , the penalty coefﬁcient σk is increased
and the resultant unconstrained problem is solved. This solution is taken as the initial guess
for the next iteration. Solutions of the successive unconstrained problems will converge to the
solution of the original constrained problem.
Larger σk values lead to ill-conditioned unconstrained subproblems.
3.3.2 Augmented Lagrangian method
Augmented Lagrangian methods try to avoid the disadvantage of penalty algorithms. They have
similarities to penalty methods in that they replace a constrained optimization problem by a
series of unconstrained problems; the difference is that the augmented Lagrangian method adds
an additional term to the unconstrained objective. This additional term is designed to mimic a
Lagrange multiplier.
The constrained optimization problem in 3.2 is reformulated as
min Φk(x) = f (x)+
µk
2
ci (x)2
− λi ci (x) (3.4)
After each iteration. in addition to updating µk, λ is updated according to the rule
λi+1 = λi −µkci (x) (3.5)

where xk is the solution to the unconstrained problem at the kth
step.
The variable λ is an estimate of the Lagrange multiplier, and the accuracy of this estimate im-
proves at every step. The major advantage of the method is that unlike the penalty method,
it is not necessary to take µ → ∞ in order to solve the original constrained problem. Instead,
because of the presence of the Lagrange multiplier term, µ can stay much smaller.
3.3.3 Interior point method
Also referred to as barrier methods.The interior point method was invented by John Von Neu-
mann. Von Neumann suggested a new method of linear programming, using the homogeneous
linear system of Gordan (1873) which was later popularized by Karmarkar’s algorithm [19]. The
method consists of a self-concordant barrier function used to encode the convex set. Contrary
to the simplex method, it reaches an optimal solution by traversing the interior of the feasible
region.
These methods make it practical to solve convex problems up to very large size, and they are
especially effective on second order (quadratic and SOCP) problems, where the Hessians of the
problem functions are constant. Both theoretical results and practical experience show that In-
terior Point methods require a relatively small number of iterations to reach an optimal solution,
independent of the number of variables and constraints (though the computational effort per
iteration rises with the number of variables and constraints). Interior Point methods have also
beneﬁted, more than other methods, from hardware advances,instruction caching, pipelining,
and other changes in processor architecture.
This method is more elaborately explained in 3.4
3.4 IPOPT(Interior Point OPTimizer)
IPOPT is an open-source software package for large-scale nonlinear optimization. It can be used
to address general nonlinear programming problems of the form

min
x∈Rn
f (x) (3.6a)
s.t.gL
≤ g(x) ≤ gU
(3.6b)
xL
≤ x ≤ xU
(3.6c)
3.4.1 Algorithm
Ipopt replaces any inequality constraint in 3.6b by an equality constraint and a new bounded
slack variable (e.g.,gi (x) − si = 0 with gL
i
≤ si ≤ gU
i
). Considering a simpler case with variables
having only a lower bound of zero, the problem in 3.6 can be restated as
min
x∈Rn
f (x) s.t. c(x) = 0 (3.7a)
x ≥ 0 (3.7b)
IPOPT is based on interior point method and hence retorts to a auxiliary barrier problem for-
mulation
min
x∈Rn
ϕµ(x) = f (x)−µ
n
i=1
ln(xi ) (3.8a)
s.t. c(x) = 0 (3.8b)
In this reformulation the bound constraints in 3.7b are replaced by a logarithmic barrier term
which is added to the objective function. For a given value for the barrier parameter µ > 0, the
barrier objective function ϕµ(x) goes to infinity if any of the variables xi approach their bound
zero.This ensures that an optimal solution of 3.8 will be in the interior of the region defined by
3.7b.
The size of the barrier parameter µ does influence barrier term, and it can be proved under
standard conditions, that the optimal solutions of x(µ) of 3.8 converge to an optimal solution of
the original problem 3.7 as µ → 0. In interior point methods the solution strategy is to solve a
sequence of barrier problems 3.8, starting with a moderate value of µ (0.1) and a user-supplied

starting point. Once the problem is solved to a relaxed accuracy, the µ is decreased iteratively
using the last step solution as a starting point. This process is continued until a solution for 3.7,
or at least a point satisfying the first-order optimality conditions up to user tolerances, has been
found.
The first order optimality conditions for the barrier problem are given as
∇f (x)+∇c(x)y − z = 0 (3.9a)
c(x) = 0 (3.9b)
X Ze −µe = 0 (3.9c)
x,z ≥ 0 (3.9d)
with µ = 0,where y ∈ Rm
and z ∈ Rn
are the Lagrangian multipliers for the equality and bound
constraints, respectively. Not only the minimizers for 3.7, but also some maximizers and sad-
dle points satisfy 3.9 and that there is no guarantee that Ipopt will always converge to a local
minimizer of 3.7. However, the Hessian regularization aims to encourage the algorithm to avoid
maximizers and saddle points.
3.4.2 Interpreting the output
IPOPT displays intermediate information for each iteration of the optimization procedure, and
closes displaying statistics concerning the computational effort and finally the EXIT message,
indicating whether the optimization terminated successfully.
Tabel 3.1 explains the columns of the iteration output. The first column appended with "r"
indicates that the algorithm is currently in the restoration phase.
Each optimization iteration, would target that the objective function is going to the optimal
value, and the constraint violation and the dual infeasibility, as well as the size of the primal
search direction are going to zero in the end. As the value of barrier parameter is going to zero,
a decrease in the number listed in column 5 is observed. Each time the algorithm switches to
the restoration phase a different value of µ might be chosen. Larger step sizes in columns 8 and
9 indicates progress in the optimal direction. A nonzero value of δx indicates that the iterates
seem to be in a region where the original problem is not strictly convex.Nonzero values at the
end of the optimization, might indicate that the algorithm terminated at a critical point that is

not a minimizer (but still satisﬁes 3.9 up to some tolerance).
colNum header signiﬁcance
1 iter iteration counter k (r: restoration phase)
2 objective current value of objective function, f (xk)
3 inf_pr current primal infeasibility (max-norm), c(xk) ∞
4 inf_du current dual infeasibility (max-norm), 3.9a ∞
5 lg(mu) log10 of current barrier parameter µ
6 d max-norm of the primal search direction ∆xk ∞
7 lg(rg) log10 of Hessian perturbation δx
8 alpha_du dual step size αz,max
k
9 alpha_pr primal step size αx
k
10 ls number of backtracking steps l + 1
Table 3.1: IPOPT iteration output

Chapter 4
Shakedown Theory
4.1 Material Failure
The common loading phenomenon that most of the structures undergo are cyclic in nature. The
study of the material behavior under such loading conditions to understand the failure modes
like ratcheting and shakedown is of paramount importance. Examples of structures, operating
under such loading conditions may be found in mechanical engineering,like pressure vessels,
aircraft gas propulsion engines, general machinery. In civil engineering such situations arise in
constructions like dams, pavements, offshore platforms, buildings and bridges under seismic
actions.
The structures subjected to cyclic loads, capable of inducing plastic strains exhibit a short term
transient response followed by a steady state response. The steady state response can be cate-
gorized into three generic types.
• Elastic shakedown :For relative low load amplitudes, the structure shakes down elastically,
i.e. the behavior appears to be purely elastic.
• Plastic shakedown : For certain patterns and levels of loading, plastic strain increments
appear to be alternating in sign over the cycle and tend to cancel each other, thus the total
deformation remains low. This phenomenon is called alternating or reverse plasticity and
failure may occur due to low-cycle fatigue
• Ratcheting :For certain patterns and levels of loading, the plastic strain increments in each
load cycle are of the same sign resulting to total strains and thus displacements to be large
22

CHAPTER 4. SHAKEDOWN THEORY 23
so that the structure becomes unserviceable. This situation is called incremental collapse
or ratcheting.
Figure 4.1: Elastic Shakedown Figure 4.2: Alternating Plasticity
Figure 4.3: Ratcheting
4.2 Shakedown
The shakedown of a structure indicates that the damage stops evolving after a ﬁnite number of
loading cycles. This is due to the fact that a stationary self-equilibrium stress ﬁeld is formed

and the total dissipated energy becomes stationary. Therefore, the prediction of shakedown or
collapse of structures under variable repeated loads is very important and useful for structural
design and safety assessment.
The shakedown static and kinematic theorems have been constructed for the elastic perfectly
plastic materials by Melan [28] and Koiter [20]. The basic concept of the theorems is their path
independence: The theorems determine the time-independent boundary in the loading space,
under which a structure is safe regardless of particular loading histories, while the structure fails
if the boundary is violated unrestrictedly. This ensures that, one does not need to follow an exact
loading history to solve a shakedown problem by the usual direct incremental analysis, and that
simplifies the work. Moreover, with any trial admissible static field the static theorem gives a
lower bound on the shakedown limit, while with a trial admissible kinematic field the kinematic
theorem gives an upper bound.
4.3 Kinematic Shakedown theorem
Also referred to as the Upper bound theorem was introduced by Koiter. The theorem is a math-
ematical formulation of the shakedown problem using the plastic strain fields.
For an elastic-plastic body, corresponding to a cycle of the loading regime with displacement
field ∆u an admissible cycle of plastic strain ∆εp
is introduced. The plastic strain rate ˙εp
need
not necessarily be compatible at each instant of the time cycle but the plastic strain accumula-
tion over the cycle (4.1) must be kinetically compatible
∆εp
=
T
0
˙εp
dt (4.1)
such that



∆ε
p
i j
=
1
2
∂∆ui
∂xj
+
∂∆uj
∂xi
in Ω
∆ui = 0 onΩu
(4.2)
The upper bound theorem reads:If any kinematically admissible mechanism of plastic collapse
can be found in which the rate of work done by the elastic stresses due to the load exceeds
the rate of plastic dissipation, then incremental collapse will take place. The ratio of the work
done by the elastic stresses to the plastic dissipation has a maximum in the “true” mechanism

of collapse, so that any other mechanism gives upper bound to the shakedown limit.
Mathematically
• The structure will shakedown íf
α
T
0
dt
Ω
σE
i j (x,t)˙ε
p
i j
dΩ ≤
T
0
dt
Ω
Dp
(˙ε
p
i j
)dΩ (4.3)
• The structure will not shakedown íf
α
T
0
dt
Ω
σE
i j (x,t)˙ε
p
i j
dΩ >
T
0
dt
Ω
Dp
(˙ε
p
i j
)dΩ (4.4)
Where α is the shakedown load factor
4.4 Static Shakedown theorem
Melan theorem states that for an elastic-plastic medium, the shakedown state is achieved only
when a positive factor α > 1 and a time-independent residual stress ﬁeld ρ can be found, whose
superposition with σE
does not exceeds the yield limit at any given time t and any given point x.
F(ασE
(x,t)+ρ(x),σY ) ≤ 0 (4.5)
∇.ρ(x,t) = 0 in Ω, (4.6a)
σE
(x,t).n = t on ∂Ωt , (4.6b)
ρ.n = 0 on ∂Ωt (4.6c)
where n is surface normal, Ωt is the static boundary, and t the external loading
For von Mises yield criterion, function F becomes
F(σ,σY ) =
3
2
σ : σ −σY (4.7)

4.5 Proof of theorem
In order to prove the theorem we consider a elastic-perfectly plastic solid with Von-Mises yield
surface, associated flow law, and uni-axial tensile yield stress Y. Below assumptions are made
• The displacement u=0 on part of the boundary of the solid ∂1V
• The remainder of the boundary ∂2V is subjected to traction t(t).
The following terms are defined
1. Let [ui j ,εi j ,σi j ] denote the actual history of displacement, strain and stress induced in
the solid by the applied loading. The strain can be resolved into elastic and plastic com-
ponents εi j = εe
i j
+ε
p
i j
2. Let [uE
i j
,εE
i j
,σE
i j
] denote the history of displacement, strain and stress induced by the pre-
scribed traction in a perfectly elastic solid with identical geometry.
3. We introduce (time dependent) residual stress ρi j and residual strain γi j fields satisfy
σi j = σE
i j +ρi j
εi j = εe
i j +ε
p
i j
= εE
i j +γi j +ε
p
i j
(4.8)
To prove the Lower bound shakedown theorem we consider the strain energy W associated with
the difference between the time dependent residual stress field ρi j and the time independent
residual stress field ¯ρi j
W =
1
2 V
(ρi j − ¯ρi j )C(−1)
i jkl
(ρkl − ¯ρkl )dV ≥ 0 (4.9)
In this equation ρ(x,t) is the actual time-dependent residual stress field and ¯ρ(x,t) represents
the time-independent residual stress field. C(−1)
i jkl
is the elastic compliance tensor.
The time derivative of 4.9 can be calculated as
˙W =
V
Ci jkl (ρi j − ¯ρi j )
dρkl
dt
dV ≥ 0 (4.10)
The product Ci jkl ρkl = γi j . Using Equation 4.8

˙W = −
V
(ρi j − ¯ρi j )
dε
p
i j
dt
dV +
V
dεe
i j
dt
−
dεE
i j
dt
dV ≥ 0 (4.11)
Using the principal of virtual work, the second integral can be expressed as an integral over the
boundary of the solid
V
(ρi j − ¯ρi j )nj
dεe
i j
dt
−
dεE
i j
dt
dV = 0
since (ρi j − ¯ρi j )nj = 0 on ∂2V and ˙ui − ˙uE
i
=0 on ∂1V
Using the results Equation 4.11 can be written as
˙W = −
V
dε
p
i j
dt
dV = −
V
σi j −(σE
i j + ¯ρi j )
dε
p
i j
dt
dV ≥ 0 (4.12)
The stress σi j corresponding to the plastic strain rate ˙ε
p
i j
is always greater than the (σE
i j
+ ¯ρi j ).
The inequality ˙W ≥ 0 is satisﬁed if the plastic strain vanishes or σi j −(σE
i j
+ ¯ρi j ) = 0. In either
case the solid will shakedown to the elastic state.
4.6 Von mises yield criterion
According to the von Mises’s theory, a ductile solid will yield when the strain energy density
reaches a critical value for that material.Strain energy density, W, has units of Energy / Volume
and is
W = σ : dε (4.13)
For linear elastic materials it is
W =
1
2
σ : dε =
1
2
[σxxεxx +σyy εyy +σzzεzz +2(σxy εxy +σyzεyz +σxzεxz)] (4.14)
Since σ = σhyd +σ and ε = εhyd +ε
W =
1
2
σhyd : εhyd +
1
2
σ : ε (4.15)
The von Mises stress is directly related to the deviatoric strain energy term of 4.15
Using Hooke’s law

W =
1
2
σ : ε =
1
4G
σ : σ (4.16)
Using a effective stress that would be proportional to the von mises stress.
W =
1
4G
(σEq)2
(4.17)
Comparing equations 4.17 and 4.16
σEq = σ : σ (4.18)
Since this should be true for uni-axial stress state also, the critical value of the strain energy can
be estimated from the uni-axial test. At the instance of yielding in a uni-axial tensile test, the
state of stress in terms of principal stress is given by:
σ =





σ 0 0
0 0 0
0 0 0





the deviatoric stress tensor is
σ =





2
3
σ 0 0
0 −σ
3
0
0 −σ
3
0





Using the deviatoric stress tensor
σEq =
2
3
σ (4.19)
Scaling the equivalent stress we need to multiply 4.19 with 3
2 .
σV M =
3
2
σ : σ (4.20)

Figure 4.4: Von mises yield criterion
4.7 Loading Domain
Considering NL linearly independent varying loads. Let Pi (x,t) be the vectors spanning an NL
dimensional loading space. Pi can be split into two parts including a varying magnitude scalar
µi and time invariant base vector ˆPi . This decomposition facilitates the representation of any
loading history as a trajectory in the loading space L NL
as shown in Figure 4.5
Figure 4.5: Loading polyhedron in a three dimensional loading space

H(x,t) =
NL
i=1
Pi (x,t) =
NL
i=1
µi (x,t) ˆPi (x) (4.21)
Due to linearity σE
can be analogously decomposed
σE
(x,t) = σE
NL
i=1
Pi (x,t) =
NL
i=1
µi (x,t).σE
( ˆPi (x)) (4.22)
This representation reflects a one-to-one mapping between the load space L NL
and the Eu-
clidean space RNL
. This makes the physical nature of a specific loading insignificant, and load-
ings differ only by their magnitudes. By varying µi within the interval [µ−
i
,µ+
i
] all admissible
loads can be expressed. This interval constitutes NC = 2NL
vertices’s of a load polyhedron.
König [21] proved that ’if a given structure shakes down over any load path contained within the
boundary δΩ of a given load domain Ω then it shakes down also over any load path contained
within Ω provided that Ω is bounded’. The domain Ω is usually a convex hyper-polyhedron
defined by NC vertices’s.Therefore, it is sufficient to evaluate only the corner points of the poly-
hedron enveloping the loading domain to satisfy the shakedown condition for all possible paths
inside the loading domain. This would eliminate the time factor from Equation 4.5 resulting in
F(ασE
(ˆPk)+ρ(x),σY ) ≤ 0, ∀k ∈ 1...NC (4.23)
The reformulated numerical shakedown problem would be
PORI max.α, (4.24a)
s.t.[C]{ρ} = {0} (4.24b)
F(ασE
i (ˆPk)+ρi ,σY
i ) ≤ 0, ∀i ∈ [1,NGS],k ∈ [1,NC] (4.24c)
F(ρ) ≤ 0 (4.24d)
where NGS is the total number of gauss points in the model, and C is the self-equilibrium matrix
defined as
[C] =
NGS
m=1
wm|Jm|[Bm]T
(4.25)
where matrix B in 4.25 is the strain matrix which consists of spatial derivatives of shape func-

tions and it maps displacements into strains, w the weight factor of integration points, and J the
determinant of the Jacobian matrix. For FE models have NK nodes and NDoF degree of free-
doms for each node, it entails C ∈ RNDoF.NK x6NGS
where NDoF equals 3 for 3D case, and 2 for
2D case.

Chapter 5
Damage
Damage, in metals is the process of creation and growth of micro-voids and micro-cracks. These
discontinuities lead to a significant changes in the properties of the material which is rather
continuous in the meso scale level. The continuous media is generally referred to as RVE with
properties represented by homogenized variables. The voids are small at the RVE level but large
enough to be considered at the micro level.
5.1 Damage representation
The simplest way to define a damage variable would be the volume density ratio of the mi-
crovoids.
Dv =
δVvoids
δVRV E
(5.1)
For generality where both micro-cavities and micro-cracks may exist, the damage variable can
defined by the surface density of micro-cracks and intersections of micro-voids lying on a plane
cutting the RVE of cross section δS (Figure 5.1 ). For the plane with normal n where this density
is maximum, we have
D(n) =
δSD
δS
(5.2)
32

CHAPTER 5. DAMAGE 33
Figure 5.1: Physical damage and mathematical continuous damage [25]
5.2 Effective Stress
In the case of isotropic damage, the scalar variable D(n) is not dependent on the surface normal.
The damage variable is a scalar.
D =
δSD
δS
(5.3)
It can be used as such for one-dimensional problems. It can also be used as an easy evaluation
of the approximate damage in three-dimensional problems,particularly in case of proportional
loading.
For isotropic damage, the concept of stress acting on the resisting area (δ ˜S = δS − δSD) comes
into picture.
• For uni-axial isotropic damage this mean value of the micro-stresses is given by the force
equilibrium [31]
˜σδ ˜S = σδS (5.4)
D =
δSD
δS
=
δS −δ ˜S
δS
(5.5)

˜σ =
σ
1−D
(5.6)
• For multi-axial isotropic damage, all the stress components are considered. The effective
stress tensor is given by
˜σi j =
σi j
1−D
(5.7)
The damage variable D can be considered as an internal state variable characterizing the irre-
versible deterioration of a material in the thermodynamical sense.
In Equation 5.6 the value of D=0 corresponds to undamaged state, D ∈ (0,Dc) corresponds to a
partly damaged state and D = Dc deﬁnes the complete local rupture (Dc ∈ [0,1]).
5.3 Constitutive relations
In the physical sense, material degradation is a cumulative effect of initiation, growth and coa-
lescence of micro-cracks or micro-voids induced by large plastic strains. The phenomenon of
ductile plastic damage leads to plastic fracture. The evolution of isotropic ductile plastic dam-
age is modeled by Lemaitre [26]. The differential (5.8) and integral forms (5.9) are
˙D =
Dc
εR −εD
2
3
(1+ν)+3(1−2ν)
σH
σeq
2
˙p (5.8)
and
D =
Dc
εR −εD
2
3
(1+ν)+3(1−2ν)
σH
σeq
2
p −εD (5.9)
with



〈x〉 = x for x ≥ 0
〈x〉 = 0 for x ≤ 0
where p = 2
3
εp
: εp
1/2
is the accumulated plastic strain, σeq = 3
2
σD
: σD
1/2
represents the
von Mises equivalent stress, σD
= (σ − σH I) is the deviatoric stress, σH = 1
3
tr(σ) the hydro-
static stress. The material constants εD (damage threshold strains), εR (strain at fracture) and

Dc(critical value of damage parameter at fracture) inﬂuence the model. These material con-
stants are determined experimentally.
5.4 Damage parameter
In order to control the degree of material damage, local bounds on the evolution of the damage
parameter D have to be imposed. As damage is basically generated by plastic strains, bounds
for damage can be given in this special case by bounding of the equivalent plastic strain.
The yield function FL coupled with damage is derived from the classical von Mises criterion
FL =
3
2
σD
1−D
:
σD
1−D
1/2
−σS ≤ 0 (5.10)
where σS is the limit yield stress. The evolution of plastic strains with damage being taken into
account then becomes
˙εp
= ˙λ
∂FL
∂σ
=
3
2
˙λ
1−D
˜σD
˜σeq
(5.11)
where ˙λ is the plastic multiplier deﬁned in the usual sense. The equivalent plastic strain rate is
given by
˙p =
2
3
˙εp : ˙εp =
˙λ
1−D
(5.12)
From 5.21 and 5.24
σeq ˙p = σ : ˙εp
(5.13)
such that
σeq ˙p =



0 for σeq ≤ (1−D)σS
(1−D)σS ˙p for σeq = (1−D)σS
(5.14)
If the yield surface contains the origin of the coordinate system and the material is initially virgin
then there exist two constants ζ and ξ such that
ζ(˙εp
: ˙εp
)(1/2)
≤ σ : ˙εp
≤ ξ(˙εp
: ˙εp
)(1/2)
(5.15)

where ζ = . 2
3(1−DcσS and ξ = 2
3σS.
For initially virgin material, according to Dorosz and König [? ] we have
˙εp
: ˙εp
)(1/2)
≤
t
0
˙εp
: ˙εp
)(1/2)
dt (5.16)
or
p ≤
t
0
˙pdt (5.17)
This allows to bound locally the accumulated plastic strain by
p ≤
1
(1−Dc)σS
t
0
σ : ˙εp
dt (5.18)
If shakedown is ensured with the safety factor α > 1, the total dissipated energy can be bounded
t
0
V
0
σ : ˙εp
dV dt ≤
α
α−1 V
1
2
{ρ} : {E}−1
: {ρ}T
dV (5.19)
and for the accumulated plastic strain one gets
p ≤
1
V
1
(1−Dc)σS
α
α−1 V
1
2
{ρ} : {E}−1
: {ρ}T
dV (5.20)
For a body B of volume V with a sufﬁciently smooth surface S ,the upper bound of the ductile
damage parameter for any sub domain of B is modeled by Hachemi [13] and is expressed as
D =
Dc
εR −εD
R
V
1
(1−Dc)σS
α
α−1 V
1
2
{ρ} : {E}−1
: {ρ}T
dV −εD (5.21)
where R is deﬁned by R = 2
3
(1+ν)+3(1−2ν) σH
σeq
2
, with D satisfying the condition 0 ≤ D ≤ Dc
for any point on the body B. The compliance tensor E−1
in 5.21 is calculated from the modulus
E and poisson ratio µ
C =














1
E
−
µ
E
−
µ
E
0 0 0
−
µ
E
1
E
−
µ
E
0 0 0
−
µ
E
−
µ
E
1
E
0 0 0
0 0 0
2µ+2
E 0 0
0 0 0 0
2µ+2
E 0
0 0 0 0 0
2µ+2
E














(5.22)

Equation 5.21 is simplified for the implementation in MATLAB. The volume integral [V ] is re-
placed with a summation over the gauss integration points of the finite element model.
D =
Dc
εR −εD
R
V
1
(1−Dc)σS
α
α−1
1
2
NGS
i=1
{ρ} : {C} : {ρ}T
∗Gaussvol −εD (5.23)
Considering the damage effects into the shakedown problem leads to additional set of inequality
constraints involving the Damage parameter defined in Equation5.21
D(ρ,α)−Dc < 0 (5.24)
The reformulated shakedown optimization problem
PD max.α, (5.25a)
s.t. : [C]{ρ} = {0} (5.25b)
F(ασE
i (ˆPk)+ρi ,(1−D(ρ,α)σY
i )) ≤ 0, ∀i ∈ [1,NGS],k ∈ [1,NC] (5.25c)
F
ρ
1−D
< 0 (5.25d)
D(ρ,α)−Dc < 0 (5.25e)
5.5 Damage curve fitting
The damage parameter from Equation 5.21 is modeled using the curve fitting method of MAT-
LAB. An exponential curve fitting is done using the genetic algorithm.
Substituting the material constants εD = 0.02 (damage threshold strains), εR = 0.37 (strain at
fracture) and Dc = 0.24(critical value of damage parameter at fracture) into 5.23 we have
D(x) =
Dc
εR −εD
(x) =
0.24
0.35
(x) (5.26)
where x is
R
V
1
(1−Dc)σS
α
α−1
1
2
NGS
i=1
{ρ} : {C} : {ρ}T
∗Gaussvol −εD (5.27)

Figure 5.2: Damage function fitting
D = a.ebx
−c (5.28)
According to the constraint specified in Equation 5.24 the maximum value that x can take is
equal to εR −εD.
Sampling the function x as specified in Equation 5.27 using a exponential function, a curve fit-
ting is done to obtain the constants of the function.
In the Figure 5.2 the Equations 5.26, 5.28 are plotted as Original function and Exponential func-
tion. The coefficients of Equation 5.28 are obtained by the genetic algorithm.

Chapter 6
MATLAB implementation
6.1 MATLAB Interface for IPOPT
In order to solve the non linear optimization problem using IPOPT, one can choose among many
available programming interfaces like Java,C++, Fortran, AMPL and MATLAB. The MATLAB in-
terface to IPOPT uses the mex interface of MATLAB.
6.1.1 IPOPT function call
The function call to the IPOPT solver from MATLAB is
[x,in f o] = ipopt(x0, f uncs,options)
The function arguments and return values for IPOPT strcuture are
• x0 : Vector of initial values for the primal variables, length N.
• funcs : A struct with the fields Column 1 Table 6.1
• options : A struct with the fields Column 2 Table 6.1
• x : optimized values of the primal variables.
• info : A struct with the fields A struct with the fields Column 3 Table 6.1
39

CHAPTER 6. MATLAB IMPLEMENTATION 40
funcs. options. info.
objective lb zl
gradient ub zu
constraints cl lambda
jacobian cu status
jacobianstructure auxdata iter
hessian zl cpu
hessianstructure zu
iterfunc lambda
ipopt
Table 6.1: MATLAB function struct
6.1.2 funcs MATLAB struct
A struct of MATLAB function handles passed to the IPOPT solver. The standard MATLAB func-
tion signature
function retV al = f uncName(ar g1,ar g2)
• objective(x, auxdata) : Function handle for the objective function. Evaluates the function
value at the current point x.
• gradient(x, auxdata) : Returns the gradient (vector of partial derivatives w.r.t. x) of the
objective at the current point x.
• constraints(x, auxdata) : Returns a vector of length M containing the values of the con-
straints at the current point x. Its size has to be equal to the lower and upper limits in
options.cl and options.cu.
• jacobian(x) : Returns a M×N matrix in MATLAB’s sparse format. It contains all ﬁrst deriva-
tives of the constraints w.r.t. the primal variables at the current point x
• jacobianstructure() : Returns a sparse M×N matrix with ones wherever the constraint Ja-
cobian potentially has nonzero entries.
• hessian : Returns the N×N Hessian matrix, i.e. second partial derivatives, of the Lagrangian
w.r.t. the primal variables x.
• hessianstructure : Returns a sparse, lower-triangular N×N matrix with ones wherever the
lower-triangular part of the Hessian of the Lagrangian potentially has nonzero entries.

CHAPTER 6. MATLAB IMPLEMENTATION 41
6.1.3 options MATLAB struct
• lb : Lower bounds for the primal variables x, vector of length N.
• ub : Upper bounds for the primal variables x, vector of length N
• cl / cu : Lower / upper bounds for the constraints, vectors of length M.
• zl / zu / lambda : Set initial values for the Lagrange multipliers for the lower / upper bounds
on the variables and for the constraints
• ipopt : This field is used to set all parameters for IPOPT
6.2 Symbolic Toolbox
Mathematical optimization involves extensive evaluations function derivatives for the calcula-
tion of the Jacobian and Hessian. The shakedown problem introduced in the earlier chapters
(4) involves complex mathematical functions whose derivatives have to be calculated at each
iteration step of the optimization.
MATLAB symbolic math toolbox provides functions for solving and manipulating symbolic
math expressions and performing variable-precision arithmetic. It enables analytical evalua-
tions of differentiation, integration, simplification, transforms, and equation solving.
The mathematical formulation of the shakedown problem (4.24 ) is implemented in MATLAB
using the symbolic toolbox.For this purpose MATLAB functions are defined according to Section
6.1.
The shakedown objective function is a function of the residual stress field ρ and α. The function
is represented as a vector of length N, where N is 6NGS+1.
PORI (ρ,α)
and the objective if to maximize the load factor α subjected to the constraints (4.24)
At each optimization step the objective function is evaluated iteratively at each gauss integration
point of the finite element model with NE elements.
The function handles are generated and saved for the symbolic MATLAB equations created for
the shakedown problem.Below are the function evaluations for the shakedown problem (4.24).
The vectorial representations can be found in the appendix at the end of the document.

Chapter 7
Validation of the developed numerical
method
In the earlier sections emphasize is laid on introducing and formulating the shakedown prob-
lem. These bounding techniques for calculating shakedown loads are of great importance as
design criteria since these eliminate the need for performing full cyclic loading programs either
numerically or experimentally.
In order to demonstrate and validate the proposed optimization technique using the Ipopt
solver, problem of plate with a circular hole under various combinations is considered. The
3D structure is discretized using ABAQUS 6.13 by the method of finite elements. The FE tool is
capable of graphically representing the model , the definition of boundary conditions, the finite
element calculation and the subsequent graphical and tabular presentation of the results.
The optimization problem is solved using the Ipopt, and the result is validated using the Gurobi
optimization algorithm. Gurobi Optimizer is a state-of-the-art solver for mathematical pro-
gramming.
7.1 Square plate with a hole under a biaxial load system
A biaxially loaded 3D square plate with a hole as shown in Figure is used 7.1
For validation purpose a simple 3D structure is modeled and discretized using ABAQUS. Using
symmetry boundary conditions only a quarter of the structure is analyzed. The discretized mesh
is as shown in the Figure 7.2
42

CHAPTER 7. VALIDATION OF THE DEVELOPED NUMERICAL METHOD 43
Figure 7.1: Square plate with a hole under a biaxial load system
Length L (mm) Hole DiameterD(mm) Thickness h(mm) D/L
100 20 2 0.2
Table 7.1: Geometry properties
7.2 Elastic Analysis
For the purpose of shakedown analysis an elastic stress ﬁeld for the structure has to be obtained.
This is done using the pure elastic material properties in ABAQUS and simulating it under dif-
ferent loading conditions. The geometric and material properties used in the FE model are tab-
ulated in tables 7.1 and 7.2
Young’s Modulus E (MPa) Poisson Ratio ν Yield stress σY (MPa)
210000 0.3 280
Table 7.2: Material Parameters

Z
Y
X
Figure 7.2: Finite element discretization
7.2.1 Pressure Px=100 MPa
Z
Y
X
Figure 7.3: Pressure Px=100 in x-direction

(Avg: 75%)
S, Mises
+2.970e+01
+7.206e+01
+1.144e+02
+1.568e+02
+1.991e+02
+2.415e+02
+2.838e+02
+3.262e+02
Figure 7.4: von Mises stress with Px=100
7.2.2 Pressure Py=100 MPa
Z
Y
X
Figure 7.5: Pressure Py =100 in y-direction

(Avg: 75%)
S, Mises
+2.804e+01
+7.057e+01
+1.131e+02
+1.556e+02
+1.982e+02
+2.407e+02
+2.832e+02
+3.258e+02
Figure 7.6: von Mises stress with Py =100
7.2.3 Pressure in x,y directions
Z
Y
X
Figure 7.7: Pressure in x- and y- direction with Py =Py =100 MPa

(Avg: 75%)
S, Mises
+9.984e+01
+1.147e+02
+1.295e+02
+1.443e+02
+1.591e+02
+1.739e+02
+1.887e+02
+2.035e+02
Figure 7.8: von Mises stress with Py =Py =100 MPa
7.3 Limit load with Incremental Analysis
For this purpose a quarter of the plate is modeled using Hypermesh. A ABAQUS template is
used. A three dimensional, 8 node mesh element is used (C3D8). The mesh data is exported into
ABAQUS format for solving it using the incremental method. A elastic-perfectly plastic material
model with σy =280 MPa is used.
A pressure Px=280 MPa is applied incrementally. ABAQUS applies this load over a time period
of 1 second with auto time stepping. The solution fails to converge after a time frequency of
0.8070. This factor multiplied with the applied load gives the limit load for the structure.
Px(limit) = Px ∗αL = 225.96MPa
where αL =0.807 is the limit load factor.
7.4 Direct Method
For the direct method a loading domain L with two load vertices’s in used. Arbitrary orthogonal
loads P1
x and P2
y (7.9)are applied alternately on a reference pure elastic RVE to calculate the stress
ﬁelds σE
1 and σE
2 respectively. With the introduction of an angle θ, a combined loading ˆP1 is
obtained. This represents the cumulative effect of proportionally varied loads P1
x and P2
y .

Figure 7.9: Elastic Stress superposition
In order to calculate the limit load of the structure, ˆP1 representing the maximum load and ˆP2
that represents the unloaded condition are used together in the optimization problem (4.24).
The direct method is implemented using different formulations of the problem using Gurobi
and Ipopt optimization algorithms. The limit load factor (αL) and shakedown load factor (αSD)
for various combinations of θ are tabulated in Table 8.2
αL αSD
θ Gurobi Ipopt Gurobi Ipopt
0° 2.259 2.259 1.767 1.767
10° 2.446 2.446 1.918 1.918
20° 2.686 2.686 2.169 2.169
30° 2.976 2.976 2.588 2.588
40° 3.356 3.356 3.330 3.330
50° 3.354 3.354 3.334 3.334
60° 2.979 2.979 2.591 2.591
70° 2.685 2.685 2.172 2.172
80° 2.442 2.442 1.921 1.921
90° 2.259 2.259 1.767 1.767
Table 7.3: Comparison between Gurobi and Ipopt

0 0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
2
2.5
3
Px/σx
→
Py/σy
→
Ipopt αL
Ipopt αSD
Gurobi αL
Gurobi αSD
Figure 7.10: Graphical plot for the various Limit and Shakedown load factors
7.5 Considering the Damage of material
The plate with a hole model is also used to demonstrate the reformulated shakedown problem
5.25 which includes the damage parameter as an additional constraint. The resulting load fac-
tors for various combinations of angles is tabulated.

θ αDam αSD
0° 1.767 1.755
10° 1.918 1.905
20° 2.169 2.155
30° 2.588 2.570
40° 3.330 3.308
50° 3.334 3.312
60° 2.591 2.574
70° 2.172 2.158
80° 1.921 1.908
90° 1.767 1.757
7.6 Residual stress field visualization
The optimized residual stress at each gauss point for the model is obtained from Ipopt is used to
calculate the effective stress. The effective stress is visualized using the Temperature predefined
field in ABAQUS.
Figure 7.11: Visualization as a temperature field in ABAQUS (Limit Analysis)

Figure 7.12: Visualization as a temperature ﬁeld in ABAQUS (Shakedown)
Figure 7.13: Visualization as a temperature ﬁeld in ABAQUS (Damage)

Chapter 8
Case Studies
The validated Optimization model is used to analyze the real micro-structures. For this purpose
a 30*30 µm models are used. The entire process of generating a FE mesh using the SEM image
is automated using the MATLAB and Python scripting capabilities of ABAQUS. The following
section deals with the technique used for handling the real micro structures.
8.1 Image Recognition and mesh generation
8.1.1 Image Toolbox: MATLAB
Image Processing Toolbox [19] provides a comprehensive set of reference-standard algorithms
and graphical tools for image processing, analysis, visualization, and algorithm development. It
permits operations including image enhancement, image deblurring, feature detection, noise
reduction, image segmentation, spatial transformations, and image registration.
Image Processing Toolbox supports a diverse set of image types, including high dynamic range,
giga pixel resolution, ICC-compliant color, and tomographic. Graphical tools allows the user to
explore an image, examine a region of pixels, adjust the contrast, create contours or histograms,
and manipulate regions of interest.
As discussed in the earlier sections the image analysis of a micro-structure image of a composite
may help to reduce the complexity of investigated structure. For this purpose MATLAB image
processing tool box is applied.
The primary goal of image analysis of the micro-structure is distinguish the areas that belongs
to a certain constituent. To realize purpose, several MATLAB functions for image processing are
52

CHAPTER 8. CASE STUDIES 53
used collectively.
The steps involved in image analysis are:
• Acquire SEM image and crop to size
• Clean up image and convert to binary data
8.1.2 Image clean-up
• imopen performs morphological opening on the gray scale image with the structuring el-
ement. The elements used in the code are circular element with radius taken as input from
the user. Once the image contain a radius less than the input pixels, it will be removed by
being substituted with the circular shaped structuring element.
• Store the result of the Step 1 as an image for further processing.
• graythresh computes a global threshold (level) that can be used to convert an intensity
image to a binary image with im2bw. Graythresh returns a normalized intensity value that
lies within [0, 1].
• im2bw converts the gray scale image I to a binary image. The output image replaces all
pixels in the input by luminance greater than level with 1 (white) and the others with 0
(black).
After an SEM image is processed after these steps, the result will be processed further by
ABAQUS to generate the mesh.
8.1.3 FE mesh technique: ABAQUS
The primary difficulty for modeling the WC-Co micro-structure owe to the randomness of both
grain shape and their distribution. To overcome this difficulty, an ABAQUS library is developed
to realize the automation of model generation by directly mapping the image data to the finite
element model.
There are different criteria based on which one can overlay a fine grid over an existing coarse
grid. The criteria used in conventional approaches require manual intervention. Unlike the
normal quad grid here the WC inclusion boundary nodes are segregated based on a initially fixed

length. These nodes are later used to adaptively mesh the boundary region of the inclusions with
a relatively finer mesh [20].
This technique of selectively meshing the region with fine elements has few advantages
• Local concentration of stresses at the sharp boundaries can be avoided.
• By adjusting the density of nodes surrounding a certain area, the mesh inside these re-
gions can be much further refined with advisable cost of additional memory. This benefit
eventually gives rise to a more accurate result in the areas of interest.
• By using a very coarse grid in regions of no interest, fewer computational powers will be
wasted by simulating "blank space" of no interest
• By using a refinement criterion within the simulation to determine these areas to re-fine,
no prior knowledge is necessary of the problem’s results. This permits a higher degree of
freedom to simply "set up the problem" without spending huge amounts of time "setting
up the grid"
(a) SEM image cropped to size (b) Binary Image
(c) Edge Approximation (d) Adaptive Mesh
Figure 8.1: Overview of the automated technique

Using the above technique the ﬁnite element mesh is generated for the real micro-structure.
8.2 RVE Case study
The real micro-structures are modeled as described in above sections. These ABAQUS models
are loaded proportionally in order to get the elastic stress ﬁled data. This elastic stress data is
processed using Python and MATLAB capabilities to generate the required data structure for the
IPOPT optimization program.
Model Parameters
Number of Elements 6591
Number of nodes (NK) 6718
Element type C3D6
Degree of freedom 3
Number of Gauss integration points per element 2
Table 8.1: Model Parameters
Figure 8.2: RVE mesh in ABAQUS

θ αL αSD αD
0° 45.131 39.532 38.203
30° 53.276 49.981 49.2
45° 49.070 47.580 46.771
60° 43.831 41.493 39.980
90° 37.809 29.747 29.458
120° 37.090 28.705 25.312
135° 37.541 31.750 29.233
150° 38.748 36.114 33.711
180° 44.983 37.316 36.640
Figure 8.3: RVE mesh in ABAQUS

Chapter 9
Descriptive statistics
Descriptive statistics is the term given to the analysis of data that helps describe, show or sum-
marize data in a meaningful way such that, for example, patterns might emerge from the data.
Descriptive statistics do not, however, allow us to make conclusions beyond the data we have
analyzed or reach conclusions regarding any hypotheses we might have made. They are simply
a way to describe our data.
Descriptive statistics are very important because if we simply presented our raw data it would be
hard to visualize what the data was showing, especially if there was a lot of it. Descriptive statis-
tics therefore enables us to present the data in a more meaningful way, which allows simpler
interpretation of the data.
With the help of the tools developed to handle the real micro-structures as described in the ear-
lier chapter, we have analyzed 50 micro-structures of the WC-Co alloy. The limit and shakedown
load factors of these models are calculated using Gurobi solver. To effectively present the results
of such volume, I am using the MATLAB’s correlation plots.
9.1 Correlation plot
Correlation plot creates a matrix of plots showing correlations among pairs of variables . His-
tograms of the variables appear along the matrix diagonal. scatter plots of variable pairs appear
off diagonal. The slopes of the least-squares reference lines in the scatter plots are equal to the
displayed correlation coefﬁcients.
The parameters selected for the correlation plot are as below
57

CHAPTER 9. DESCRIPTIVE STATISTICS 58
• WC Volume Fraction
• Young’s Modulus
• Poisson ratio
• Ultimate Strength
• Endurance Stregth
An effort to represent a correlation between these features of the composite is made.
Correlation Matrix
0.9 0.95 1 1.05
E(θ = 1/4π)
0.96 0.98 1 1.02
µ(θ = 1/4π)
0.8 0.9 1 1.1
Σ∞
(θ = 1/4π)
0.96 0.98 1 1.02 1.04
0.9
0.95
1
1.05
WC Vol.%
E(θ=1/4π)
0.96
0.98
1
1.02
µ(θ=1/4π)
0.8
0.9
1
1.1
Σ∞
(θ=1/4π)
0.96
0.98
1
1.02
1.04
WCVol.%
0.34 −0.77 0.94
0.34 −0.16 0.34
−0.77 −0.16 −0.78
0.94 0.34 −0.78
Figure 9.1: 40µmx40µm RVE data( ΣU
,µ, E, WC Vol %)
A correlation between the variables is ranked on a scale from 0-1, with one depicting maximum
correlation and zero minimum.
From Fig. 9.1 the following inferences can be made
• The WC volume fraction bears a string correlation with the ultimate strength and Young’s
modulus of the composite.
• A inverse correlation between WC volume fraction and Poisson ratio is observed .

• The Young’s modulus and ultimate strength also have a good linear dependence.
From Fig. 9.2 the following inferences can be made
• The WC volume fraction bears a string correlation with the Young’s modulus of the com-
posite.
• A inverse correlation between WC volume fraction and Poisson ratio is observed .
40µmx40µm 60µmx60µm
WC % ΣU
µ E WC % ΣU
µ E
WC % - 0.77 -0.77 0.94 - 0.82 -0.81 0.96
ΣU
0.77 - -0.62 0.84 0.82 - -0.77 0.86
µ -0.77 -0.62 - -0.78 -0.81 -0.77 - -0.8
E 0.94 0.84 -0.78 - 0.96 0.86 -0.8 -
Table 9.1: Variable correlation summary
40µmx40µm 60µmx60µm
WC % Σ∞
µ E WC % Σ∞
µ E
WC % - 0.34 -0.77 0.94 - 0.41 -0.81 0.96
Σ∞
0.34 - -0.16 0.34 0.41 - -0.44 0.45
µ -0.77 -0.16 - -0.78 -0.81 -0.44 - -0.8
E 0.94 0.34 -0.78 - 0.96 0.45 -0.8 -
Table 9.2: Variable correlation summary

Correlation Matrix
0.9 0.95 1 1.0
E(θ = 1/4π)
0.96 0.98 1 1.02
µ(θ = 1/4π)
0.8 0.9 1 1.1
Σ∞
(θ = 1/4π)
0.96 0.98 1 1.02 1.04
0.9
0.95
1
1.05
WC Vol.%
E(θ=1/4π)
0.96
0.98
1
1.02
µ(θ=1/4π)
0.8
0.9
1
1.1
Σ∞
(θ=1/4π)
0.96
0.98
1
1.02
1.04
WCVol.%
0.34 −0.77 0.94
0.34 −0.16 0.34
−0.77 −0.16 −0.78
0.94 0.34 −0.78
Figure 9.2: 40µmx40µm RVE data( Σ∞
,µ, E, WC Vol %)

Chapter 10
Conclusion
In my thesis emphasis is laid on following important issues.
1. Developing a efficient optimization technique to deal with lower bound shakedown prob-
lem of real micro-structures using IPOPT.
2. Validate the developed technique against commercial optimization technique results.
3. Extend the shakedown problem to consider material damage effects on the load factor.
Formulating a optimization problem for IPOPT solver was a significant task. There are many
solver interfaces available for IPOPT. Among them are AMPL, C++, C,FORTRAN, Java and MAT-
LAB. The MATLAB interface is chosen to formulate the problem. MATLAB’s ability to handle the
sparse matrix structure and its ease of use has significantly influenced the choice of interface for
IPOPT.
The formulation caters to a any structural problem, that adheres to the data structure. Care
is taken to keep the solver flexible using robust sparse matrices and symbolic math toolbox of
MATLAB. The formulation is validated using a standard Plate with a hole model. The results
were well in sync with the results from Gurobi a commercial software tool.
Upon successfully formulating the shakedown problem, the real micro-strucutres ( SEM images)
of the composite are solved for obtaining the load factor. Two RVE models have been analysed
for lower bound shakedown analysis.
The developed MATLAB interface for IPOPT can be used to solve large problems.
61

Appendix A
MATLAB Symbolic toolbox
The yield function of the total stress ﬁeld (ασE
+ρ)
{ρ} =

















ρ11
ρ22
ρ33
ρ12
ρ23
ρ13
α

















F(σ,σY ) = −σY
2
+
ρ22
2
−ρ11 +
ρ33
2
−ασ11 +
ασ22
2
+
ασ33
2
ρ22
3
−
2ρ11
3
+
ρ33
3
−
2ασ11
3
+
ασ22
3
+
ασ33
3
+
ρ11
2
−ρ22 +
ρ33
2
+
ασ11
2
−ασ22 +
ασ33
2
ρ11
3
−
2ρ22
3
+
ρ33
3
+
ασ11
3
−
2ασ22
3
+
ασ33
3
+
ρ11
2
+
ρ22
2
−ρ33 +
ασ11
2
+
ασ22
2
−ασ33
ρ11
3
+
ρ22
3
−
2ρ33
3
+
ασ11
3
+
ασ22
3
−
2ασ33
3
+2 ρ12 +ασ12
3ρ12
2
+
3ασ12
2
+2 ρ13 +ασ13
3ρ13
2
+
3ασ13
2
+2 ρ23 +ασ23
3ρ23
2
+
3ασ23
2
Jacobian of the yield function w.r.t. {ρ} to
62

APPENDIX A. MATLAB SYMBOLIC TOOLBOX 63

















2ρ11 −ρ22 −ρ33 +2ασ11 −ασ22 −ασ33
2ρ22 −ρ11 −ρ33 −ασ11 +2ασ22 −ασ33
2ρ33 −ρ22 −ρ11 −ασ11 −ασ22 +2ασ33
6ρ12 +6ασ12
6ρ23 +6ασ23
6ρ13 +6ασ13
A

















where
A = 0+2σ12
3ρ12
2
+
3ασ12
2
+2σ13
3ρ13
2
+
3ασ13
2
+2σ23
3ρ23
2
+
3ασ23
2
+
σ11
3
+
σ22
3
−
2σ33
3
ρ11
2
+
ρ22
2
−ρ33 +
ασ11
2
+
ασ22
2
−ασ33
+
σ11
3
−
2σ22
3
+
σ33
3
ρ11
2
−ρ22 +
ρ33
2
+
ασ11
2
−ασ22 +
ασ33
2
+
σ22
3
−
2σ11
3
+
σ33
3
ρ22
2
−ρ11 +
ρ33
2
−ασ11 +
ασ22
2
+
ασ33
2
+
σ22
2
−σ11 +
σ33
2
ρ22
3
−
2ρ11
3
+
ρ33
3
−
2ασ11
3
+
ασ22
3
+
ασ33
3
+
σ11
2
−σ22 +
σ33
2
ρ11
3
−
2ρ22
3
+
ρ33
3
+
ασ11
3
−
2ασ22
3
+
ασ33
3
+
σ11
2
+
σ22
2
−σ33
ρ11
3
+
ρ22
3
−
2ρ33
3
+
ασ11
3
+
ασ22
3
−
2ασ33
3
+3σ12 ρ12 +ασ12 +3σ13 ρ13 +ασ13 +3σ23 ρ23 +ασ23
Hessian of the yield function


















2 −1 −1 0 0 0 2σ11 −σ22 −σ33
−1 2 −1 0 0 0 2σ22 −σ11 −σ33
−1 −1 2 0 0 0 2σ33 −σ22 −σ11
0 0 0 6 0 0 6σ12
0 0 0 0 6 0 6σ23
0 0 0 0 0 6 6σ13
2σ11 −σ22 −σ33 2σ22 −σ11 −σ33 2σ33 −σ22 −σ11 6σ12 6σ23 6σ13 A

















where
A = 0+6σ12
2
+6σ13
2
+6σ23
2
+2
σ22
2
−σ11 +
σ33
2
σ22
3
−
2σ11
3
+
σ33
3
+2
σ11
2
−σ22 +
σ33
2
σ11
3
−
2σ22
3
+
σ33
3
+2
σ11
2
+
σ22
2
−σ33
σ11
3
+
σ22
3
−
2sigma33
3
Yield function of the residual stress ﬁeld {ρ}
F =0+3ρ12
2
+3ρ13
2
+3ρ23
2
−YSt2
+
ρ22
2
−ρ11 +
ρ33
2
ρ22
3
−
2ρ11
3
+
ρ33
3
+
ρ11
2
−ρ22 +
ρ33
2
ρ11
3
−
2ρ22
3
+
ρ33
3
+
ρ11
2
+
ρ22
2
−ρ33
ρ11
3
+
ρ22
3
−
2ρ33
3
Jacobian of the yield function

















2ρ11 −ρ22 −ρ33
2ρ22 −ρ11 −ρ33
2ρ33 −ρ22 −ρ11
6ρ12
6ρ23
6ρ13
0

















Hessian of the yield function


















2 −1 −1 0 0 0 0
−1 2 −1 0 0 0 0
−1 −1 2 0 0 0 0
0 0 0 6 0 0 0
0 0 0 0 6 0 0
0 0 0 0 0 6 0
0 0 0 0 0 0 0


















Appendix B
IPOPT Code
The MATLAB interface is used to formulate the optimization problem for IPOPT. You can ﬁnd in
this section you can ﬁnd the code developed to achieve the objective.
1 %% Main Function call
2 function [ ldFac , simDat]= IPOPTEngine( SolverOptions , InputStruct )
3 global DataStruct IPOPTJacobian IPOPTConstraints DamageConstraint LoadFac ;
4 global CellYieldStress ;
5 IPOPTJacobian = [ ] ;
6 IPOPTConstraints = [ ] ;
7 DamageConstraint = [ ] ;
8 GenerateDataStructs ( InputStruct ) ;
9 generateGlobStructs ;
10 i f exist ( InputStruct .FunHdlName, ’ f i l e ’ ) == 2
11 load ( InputStruct .FunHdlName) ;
12 else
13 SymFunGenerator( InputStruct .FunHdlName) ;
14 end
15
16 %Intial value set for Optimization
17 i f strcmp( InputStruct . TakeGurobiInput , ’NO’ )==1
18 InitialOptiVariables (1: DataStruct . VarCount−1) = InputStruct .
InitialResidualVal ;
19 InitialOptiVariables ( DataStruct . VarCount) = InputStruct . I n i t i a l L f ;
20 else
21 load ( InputStruct . GengExtractorDat ) ;
66

APPENDIX B. IPOPT CODE 67
22 InitialOptiVariables =[ ResultSet { 1 } . ProcessedRes . ResidualStress ; ResultSet { 1 } .
objval ] ’ ;
23 end
24
25 % Bounds on variables
26 LBOptiVariables (1: DataStruct . VarCount−1) = −inf ;
27 LBOptiVariables ( DataStruct . VarCount) = 0;
28 UBOptiVariables (1: DataStruct . VarCount) = inf ;
29
30 % Equality Constraint Bounds
31 UBEqConstraints (1: DataStruct .RowCmat) =0;
32 LBEqConstraints (1: DataStruct .RowCmat) =0;
33
34 % Yield Constraint Bounds
35 UBIEqYieldConstraint (1: DataStruct .NGS) =0;
36 LBIEqYieldConstraint (1: DataStruct .NGS) =−inf ;
37
38 % Shakedown Constraint Bounds
39 UBIEqYieldShakeConstraint (1: DataStruct .NGS) =0;
40 LBIEqYieldShakeConstraint (1: DataStruct .NGS) =−inf ;
41
42 % Damage Constraint Bounds
43 UBIEqDamageConstraint (1: DataStruct . DVars) =DataStruct .D_c;
44 LBIEqDamageConstraint (1: DataStruct . DVars) =0;
45
46 UBAllConstraints = [ ] ;
47 LBAllConstraints = [ ] ;
48 InputStruct . fileName= ’ Limit ’ ;
49 i f strcmp( DataStruct . StartPointGenerate , ’YES ’ )==1
50 UBEqConstraints = [ ] ;
51 LBEqConstraints = [ ] ;
52 end
53 UBAllConstraints =[ UBAllConstraints UBEqConstraints UBIEqYieldConstraint ] ;
54 LBAllConstraints =[ LBAllConstraints LBEqConstraints LBIEqYieldConstraint ] ;
55 i f strcmp( DataStruct . ShakeDownStatus , ’YES ’ )==1
56 InputStruct . fileName= ’Shakedown ’ ;
57 UBAllConstraints =[ UBAllConstraints UBIEqYieldShakeConstraint ] ;
58 LBAllConstraints =[ LBAllConstraints LBIEqYieldShakeConstraint ] ;
59 end

60 i f strcmp( DataStruct . DamageStatus , ’YES ’ )==1
61 InputStruct . fileName=strcat ( ’Damage−’ , InputStruct . fileName ) ;
62 UBAllConstraints =[ UBAllConstraints UBIEqDamageConstraint ] ;
63 LBAllConstraints =[ LBAllConstraints LBIEqDamageConstraint ] ;
64 end
65
66 % Mandatory functions to IPOPT
67 IpoptFunStruct . objective = @IpoptHandleObjective ;
68 IpoptFunStruct . gradient = @IpoptHandleGradient ;
69 IpoptFunStruct . constraints = @IpoptHandleConstraints ;
70 IpoptFunStruct . jacobian = @IpoptHandleJacobian ;
71 IpoptFunStruct . jacobianstructure = @IpoptHandleJacStruct ;
72 IpoptFunStruct . hessian = @IpoptHandleHessian ;
73 IpoptFunStruct . hessianstructure = @IpoptHandleHessStruct ;
74
75 % Solver options
76 SolverOptions . lb = LBOptiVariables ;
77 SolverOptions .ub = UBOptiVariables ;
78 SolverOptions . cl = LBAllConstraints ;
79 SolverOptions . cu = UBAllConstraints ;
80 SolverOptions . ipopt . obj_scaling_factor = −1;
81 SolverOptions . ipopt . print_level = 5;
82 SolverOptions . ipopt . print_user_options = ’ yes ’ ;
83 SolverOptions . ipopt . print_info_string = ’ yes ’ ;
84
85
86 % Call IPOPT for optimization
87 tStart=tic ;
88 [ OptimizedVarSet , IPOPTInfo]= ipopt ( InitialOptiVariables ’ , IpoptFunStruct , SolverOptions ) ;
89 tElapsed=toc ( tStart ) /60;
90 display ( [ ’ Optimization Time (min) : ’ , num2str( tElapsed ) ] ) ;
91 display ( [ ’Optimized Load factor : ’ , num2str( OptimizedVarSet ( DataStruct . VarCount) ) ] )
;
92
93 simDat=ResultProcessing ( OptimizedVarSet , IPOPTInfo , InputStruct , SolverOptions , tElapsed )
;
94 ldFac=OptimizedVarSet ( DataStruct . VarCount) ;
95 LoadFac=OptimizedVarSet ( DataStruct . VarCount) ;

96 CellResiStress=mat2cell ( OptimizedVarSet (1:end−1) , DataStruct .StrComp . * ones( DataStruct .
NGS,1) ,1) ;
97 simDat .DamX=cellfun (@calcDamX, CellResiStress , . . .
98 CellYieldStress , DataStruct . GVol , ’UniformOutput ’ ,0) ;
99
100 %% OBJECTIVE FUNCTION AND GRADIENT EVALUATION
101 function ObjFunEval=IpoptHandleObjective ( OptimizedVarSet )
102 global DataStruct ;
103 ObjFunEval=OptimizedVarSet ( DataStruct . VarCount) ;
104
105 function g=IpoptHandleGradient ( OptimizedVarSet )
106 global DataStruct ;
107 g=[ zeros (1 , DataStruct . VarCount−1) 1];
108 g=g ’ ;
109
110 %% CONSTRAINT EVALUATION AT EACH ITERATION
111 function ConstraintEval=IpoptHandleConstraints ( OptimizedVarSet )
112 global DataStruct LoadFac MatStruct IPOPTConstraints DamageConstraint ;
113 global CellElasStress CellYieldStress ;
114
115 ConstraintEval = [ ] ;
116
117 %EQUALITY CONSTRAINTS
118 %[C] [ res ]=[0]
119
121 EqConstraint = [ ] ;
122 else
123 EqConstraint=MatStruct .MCmat*OptimizedVarSet (1:end) ;
124 end
125
126 %INEQUALITY CONSTRAINTS
127 %F( S_elastic+residual ,(1/1−D) *YSt ) )<=0
128 LoadFac=OptimizedVarSet (end) ;
NGS,1) ,1) ;
130 IEqYieldConstraint=cellfun ( @EvalYieldFun , CellElasStress , CellResiStress , . . .
132 IEqYieldConstraint=cell2mat ( IEqYieldConstraint ) ;

133
134 ConstraintEval =[ ConstraintEval EqConstraint ’ IEqYieldConstraint ’ ] ;
136 IEqYieldShakedownConstraint=cellfun ( @EvalYieldShakeFun , CellResiStress , . . .
138 IEqYieldShakedownConstraint=cell2mat ( IEqYieldShakedownConstraint ) ;
139 ConstraintEval =[ ConstraintEval IEqYieldShakedownConstraint ’ ] ;
140 end
141
142 IPOPTConstraints . Equality=EqConstraint ’ ;
143 IPOPTConstraints . InEquality=ConstraintEval ( DataStruct .RowCmat+1:end) ;
145 %INEQUALITY CONSTRAINTS
146 %D−DCritical <=0
147 DamageFun=cellfun (@EvalDamageFun, CellResiStress ( DataStruct . strtPosDam : DataStruct .
strtPosDam+DataStruct . DVars−1) , . . .
148 CellYieldStress ( DataStruct . strtPosDam : DataStruct . strtPosDam+DataStruct . DVars
−1) , . . .
149 DataStruct . GVol( DataStruct . strtPosDam : DataStruct . strtPosDam+DataStruct . DVars
−1) , ’UniformOutput ’ ,0) ;
150 IEqDamageConstraint=cell2mat (DamageFun) ;
151 ConstraintEval =[ ConstraintEval IEqDamageConstraint ’ ] ’ ;
152 DamageConstraint=IEqDamageConstraint ;
153 end
154
155 %% JACOBIAN STRUCTURE DEFINITION AND EVALUATION
156 function JacobianStruct=IpoptHandleJacStruct ()
157 global DataStruct MatStruct ;
158
159 JacobianStruct = [ ] ;
161 JacobianEqConstraint = [ ] ;
162 else
163 JacobianEqConstraint=MatStruct . MCmatStruct ;
164 end
165 JacobianStruct =[ JacobianStruct ; JacobianEqConstraint ; DataStruct .
JacobianYieldConstraint ] ;
167 JacobianStruct =[ JacobianStruct ; DataStruct . JacobianYieldShakeConstraint ] ;

168 end
170 JacobianStruct =[ JacobianStruct ; DataStruct . JacobianDamageConstraint ] ;
171 end
172
173 function JacobianEval=IpoptHandleJacobian ( OptimizedVarSet )
174 global LoadFac DataStruct MatStruct IPOPTJacobian ;
175 global CellElasStress CellYieldStress ;
176
177 LoadFac=OptimizedVarSet (end) ;
178
179 JacobianEval = [ ] ;
180
182 JacobianEqConstraint = [ ] ;
183 else
184 JacobianEqConstraint=MatStruct .MCmat;
185 end
186 JacobianEval =[ JacobianEval ; JacobianEqConstraint ] ;
187
188 % The Yield Stress Jacobian Matrix formation
NGS,1) ,1) ;
190 CellGradYield=cellfun ( @JacobianYieldFun , CellElasStress , CellResiStress , CellYieldStress
, DataStruct . GVol , ’UniformOutput ’ ,0) ;
191 MatGradResiStress=cell2mat ( CellGradYield ) ;
192 MatGradResiStress=MatGradResiStress ’ ;
193 MatGradAlpha=MatGradResiStress ( 7 : 7 : end) ;
194 MatGradResiStress ( 7 : 7 : end) = [ ] ;
195
196 GaussSet =1:1: DataStruct .NGS;
197 RowNum=kron( GaussSet , ones(1 , DataStruct .StrComp) ) ;
198 ColNum=1:1: size ( GaussSet ,2) * DataStruct .StrComp;
199 SpJacConst2Resi=sparse (RowNum,ColNum, MatGradResiStress , DataStruct .NGS, DataStruct .
VarCount) ;
200 clear RowNum ColNum;
201 RowNum=(1: DataStruct .NGS) ;
202 ColNum(1: DataStruct .NGS)=DataStruct . VarCount ;

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