Modified Secant Homogenization Method for. Particle Reinforced Metal Matrix Composites. Jorge Zahr Viñuela and José Luis Pérez Castellanos. Department of ...
A particular implementation of the Modified Secant Homogenization Method for
Particle Reinforced Metal Matrix Composites Jorge Zahr Viñuela and José Luis Pérez Castellanos Department of Continuum Mechanics and Structures Universidad Carlos III de Madrid Presented at International Conference on Composite Structures, ICCS 2013 Porto, Portugal. Update: full paper available at http://dx.doi.org/10.1016/j.compstruct.2013.09.036
TABLE OF CONTENTS 1. INTRODUCTION: CONTEXT and PREVIOUS WORK + Particulate Reinforced Metal Matrix Composites (PMMCs) + Summary of the MODIFIED SECANT METHOD for nonlinear composites. + A proposal: to use the HALPIN‐TSAI equation as the underlying required Linear Elastic Homogenization Scheme.
2. PAIRING the HALPIN‐TSAI equation with the MODIFIED SECANT METHOD + Halpin‐Tsai applied to Homogenization of elastic constants K and G. + Halpin‐Tsai applied to Homogenization of the secant constants K sec and G sec. + Resolution method.
3. APPLICATION EXAMPLE + Matrix phase with Ramberg‐Osgood hardening
TABLE OF CONTENTS 1. INTRODUCTION: CONTEXT and PREVIOUS WORK + Particulate Reinforced Metal Matrix Composites (PMMCs) + Summary of the MODIFIED SECANT METHOD for nonlinear composites. + A proposal: to use the HALPIN‐TSAI equation as the underlying required Linear Elastic Homogenization Scheme.
2. PAIRING the HALPIN‐TSAI equation with the MODIFIED SECANT METHOD + Halpin‐Tsai applied to Homogenization of elastic constants K and G. + Halpin‐Tsai applied to Homogenization of the secant constants K sec and G sec. + Resolution method.
3. APPLICATION EXAMPLES + Matrix phase with Ramberg‐Osgood hardening
1. INTRODUCTION: Context of the present study Composite materials made from:
MATRIX:
REINFORCEMENT:
•
Elastic particles (usually ceramic)
•
Non-ellipsoidal particles
•
Elastic-Plastic metal matrix with isotropic hardening
1. INTRODUCTION: Context of the present study Composite materials made from:
MATRIX:
Particulate Metal Matrix Composite (PMMC)
REINFORCEMENT:
•
Elastic particles (usually ceramic)
•
Non-ellipsoidal particles
•
Elastic-Plastic metal matrix with isotropic hardening
1. INTRODUCTION: Context of the present study
Calculation of Composite’s (or effective) Elastic Constants
• Models based on Mean Field Theory • Self Consistent methods (partially derived from M.F.T.) • Asymptotic Expansion Homogenization • Variational Limits • Computational micromechanics approaches (cell modelling)
Incomplete list of references: Eshelby, 1957 and its derivatives Kröner, 1958 ; Hill, 1964 Sánchez and Hubert and Sánchez-Palencia, 1992 Hashin-Shtrickman, 1963 Segurado and Llorca, 2002
1. INTRODUCTION: Context of the present study
Mechanical behavior of composites with nonlinear constituent phases
• Tangent Method • Secant Method (classical) • Secant Method (modified)
Incomplete list of references: Hill, 1965b Suquet, 1995 Ponte Castañeda, 1996
These methods make use of an underlying linear elastic homogenization scheme!! Usually Mean Field or Selfconsistent elastic methods are used
1. INTRODUCTION: Context of the present study MACROSCOPIC LOADING OF THE COMPOSITE, characterized by
Particles
macroscopic stress tensor
: macroscopic strain tensor :
with:
Composite
INDUCES STRESS and STRAIN STATES IN THE CONSTITUENT PHASES Matrix
p : stress tensor in particles p : strain tensor in particles with:
:
m : stress tensor in the matrix m : strain tensor in the matrix with:
F :
:
Particle volume fraction
1. INTRODUCTION: Context of the present study MACROSCOPIC LOADING OF THE COMPOSITE, characterized by
Particles
macroscopic stress tensor
: macroscopic strain tensor :
with:
Composite
INDUCES STRESS and STRAIN STATES IN THE CONSTITUENT PHASES Matrix
p : stress tensor in particles p : strain tensor in particles with:
:
m : stress tensor in the matrix m : strain tensor in the matrix with:
F :
:
Particle volume fraction
1. INTRODUCTION: Context of the present study MACROSCOPIC LOADING OF THE COMPOSITE, characterized by
Particles
macroscopic stress tensor
: macroscopic strain tensor :
with:
Composite
INDUCES STRESS and STRAIN STATES IN THE CONSTITUENT PHASES Matrix
p : stress tensor in particles p : strain tensor in particles with:
:
m : stress tensor in the matrix m : strain tensor in the matrix with:
F :
:
Particle volume fraction
1. INTRODUCTION: Context of the present study MACROSCOPIC LOADING OF THE COMPOSITE, characterized by
Particles
macroscopic stress tensor
: macroscopic strain tensor :
with:
Composite
INDUCES STRESS and STRAIN STATES IN THE CONSTITUENT PHASES Matrix
p : stress tensor in particles p : strain tensor in particles with:
:
m : stress tensor in the matrix m : strain tensor in the matrix with:
F :
:
Particle volume fraction
1. INTRODUCTION: Context of the present study Particles
CONSIDERATIONS:
• Secant compliance of particles remains constant.
1
Mp
Composite
• Secant compliance of the matrix phase increase with increasing plastic straining in this phase. Matrix
1
M sec
• This increase in matrix compliance with matrix plastic straining is governed by the Mises eq. stress,
1
M
sec m
NON LINEAR ELASTICITY
1. INTRO.: Summary of the MODIFIED SECANT METHOD for nonlinear comps. Particles
A Secant Homogenization Method applies
1
homogenization methods for composites with Linear Elastic constituent phases
Mp
Composite
to the Matrix
1
M sec
1
M
homogenization of secant properties of composites with nonlinear constituent phases
sec m
,
;
NON LINEAR ELASTICITY
1. INTRO.: Summary of the MODIFIED SECANT METHOD for nonlinear comps.
FULL SET OF EQUATIONS: ∶
Constitutive equation of MATRIX PHASE: • Compliance tensor of the matrix phase: • Bulk modulus:
• Secant shear modulus: • Equivalent plastic strain in the matrix:
̅
Note that and are the 4th order hydrostatic and deviatoric projection tensors, respectively.
∶
Constitutive equation of PARTICLES: • Compliance tensor of particles: • Bulk and shear moduli: Constitutive equation of the COMPOSITE: • Compliance tensor of Composite:
, : ,
;
1. INTRO.: Summary of the MODIFIED SECANT METHOD for nonlinear comps.
IMPORTANT MAIN ASSUMPTION IF
the CONSTITUTIVE BEHAVIOR of PARTICLES and MATRIX is ISOTROPIC AND
the DISTRIBUTION of PARTICLES within the MATRIX is GEOMETRICALLY ISOTROPIC THEN
the CONSTITUTIVE BEHAVIOR of the COMPOSITE will be also considered ISOTROPIC
Consequence:
,
;
1 3
1 2
There are only 2 secant parameters
1. INTRO.: Summary of the MODIFIED SECANT METHOD for nonlinear comps.
ORIGIN OF NON‐LINEARITY Because
(a)
and
(b)
It follows that
(c)
∶
,
;
, ;
A relation is needed between
,
↔
There are 2 options: ∶
(I) Classical Secant Method:
(II) Modified Secant Method:
∷ Our main focus !
⨂
∶ ∶
⁄
∶
∶
1. INTRO.: Summary of the MODIFIED SECANT METHOD for nonlinear comps.
PRACTICAL IMPLEMENTATION 1st
equation for the unknown
Solve the (non‐linear)
3
∶
1
:
∶
1⁄
for any given macroscopic stress tensor
2nd
Once solved for
:
• Update the secant compliance of the matrix phase: • Apply linear homogenization for the compliance of composite:
,
;
1. INTRO.: Summary of the MODIFIED SECANT METHOD for nonlinear comps.
PRACTICAL IMPLEMENTATION An elastic homogenization procedure f is required for two tasks: 1. The updating of the composite’s compliance tensor
,
2. The calculation of the derivatives in the main
;
non‐linear equation
1⁄ • For the procedure f there are SEVERAL OPTIONS • Many implementations are based in Mean Field Theory • Most them make use of the ESHELBY tensor ‘S’ • For easy of calculation of ‘S’, ellipsoidal inclusions are considered in most cases. • In our materials, particles do have edges and irregular shapes. • In our implementation, we propose to set f as the HALPIN‐TSAI equation.
TABLE OF CONTENTS 1. INTRODUCTION: CONTEXT and PREVIOUS WORK + Particulate Reinforced Metal Matrix Composites (PMMCs) + Summary of the MODIFIED SECANT METHOD for nonlinear composites. + A proposal: to use the HALPIN‐TSAI equation as the underlying required Linear Elastic Homogenization Scheme.
2. PAIRING the HALPIN‐TSAI equation with the MODIFIED SECANT METHOD + Halpin‐Tsai applied to Homogenization of elastic constants K and G. + Halpin‐Tsai applied to Homogenization of the secant constants K sec and G sec. + Resolution method.
3. APPLICATION EXAMPLE + Matrix phase with Ramberg‐Osgood hardening
2. PAIRING the HALPIN‐TSAI eq. with the MODIFIED SECANT METHOD
SOME BACKGROUNDS: Halpin‐Tsai generalities 1 1
1 where
: microstructural parameter : particle volume fraction
1.
The Halpin and Tsai (1967) model was developed (and has been frequently used) for the homogenization of composites reinforced mainly with fibers or short fibers.
2.
Halpin and Kardos (1976) showed the theoretical link of the HT equation with the self consistent homogenization scheme of Hill (1964). They provided values for the parameter in a form related to particle aspect ratio R (restricted range of R).
3.
Recently, Zahr and Pérez‐Castellanos (2011) showed that the Halpin‐Tsai equation can be used to homogenize elastic properties of Particulate Metal Matrix Composites, with particles with edges and irregular shape. We chose this to be used as the underlying linear homogenization f required for the Modified Secant Method
2. PAIRING the HALPIN‐TSAI eq. with the MODIFIED SECANT METHOD
SOME BACKGROUNDS: the approach of Zahr & Pérez Castellanos (2011) • Approach based on computational micromechanics (multi‐particle cell modeling) • Particles with edges and irregular shape were modeled as PRISMATIC PARTICLES in RANDOM or CONTROLLED ORIENTATION. • Multi‐particle Cells representing Isotropic microstructural scenarios were built and analyzed by Finite Element for a range of values of F. • Finite Element results provided data for fitting the Halpin‐Tsai ‘ ’ parameter for these microstructural scenarios (non covered previously in the literature) • This led to simple explicit expressions for homogenization of E, G and k: + Young Modulus:
1 1
+ Shear Modulus:
1 1
+ Homogenized Bulk Modulus as a derived property:
3 3
2. PAIRING the HALPIN‐TSAI eq. with the MODIFIED SECANT METHOD
SOME BACKGROUNDS: the approach of Zahr & Pérez Castellanos (2011) EXAMPLES OF CELLS ANALYZED: RANDOM ORIENTATION F = 10 % N = 40 particles 3 particle aspect ratios: R = 0.3
R = 1.0
R = 3.0
Periodic geometry Periodic boundary conditions RANDOM ORIENTATION F = 30 % N = 40 particles 3 particle aspect ratios: R = 0.3
R = 1.0
R = 3.0
Periodic geometry Periodic boundary conditions
2. PAIRING the HALPIN‐TSAI eq. with the MODIFIED SECANT METHOD
SOME BACKGROUNDS: the approach of Zahr & Pérez Castellanos (2011) EXAMPLES OF CELLS ANALYZED: RESTRICTED ORIENTATION F = 10 % N = 40 particles 3 particle aspect ratios: R = 0.3
R = 3.0
R = 1.0 (not shown)
Periodic geometry Periodic boundary conditions RESTRICTED ORIENTATION F = 30 % N = 40 particles 3 particle aspect ratios: R = 0.3
R = 3.0
R = 1.0 (not shown)
Periodic geometry Periodic boundary conditions
2. PAIRING the HALPIN‐TSAI eq. with the MODIFIED SECANT METHOD
SOME BACKGROUNDS: the approach of Zahr & Pérez Castellanos (2011) RELEVANT RESULTS Shape parameter ‘sL’ of the Halpin‐Tsai equation for microstructural scenarios with
RANDOM ORIENTATION OF PARTICLES R=1/3
R=1
R=3
E
sE = 2.05
sE = 1.70
sE = 2.00
G
sG = 2.15
sG = 1.80
sG = 2.10
Provided excellent agreement between HT predictions and experimentally measured elastic constants for a real composite (Al alloy matrix + SiC particles in F = 12%, 25% and 35%)
2. PAIRING the HALPIN‐TSAI eq. with the MODIFIED SECANT METHOD
ACTUAL PAIRING of HALPIN‐TSAI + MODIFIED SECANT METHOD 1st
equation :
We combine the non‐linear
3
∶
1
∶
1⁄
with the previously stated assumption of MACROSCOPIC ISOTROPY:
1 3
1 2
to obtain:
3 1
1⁄3 1⁄
∶ ∶
1⁄2 1⁄
∶
∶
2. PAIRING the HALPIN‐TSAI eq. with the MODIFIED SECANT METHOD
PAIRING HALPIN‐TSAI WITH THE MODIFIED SECANT METHOD 1st
Which is equivalent to
1 1
1⁄ 1⁄
“Modified measure” of the Mises stress in the matrix phase
3
1⁄ 1⁄
Macroscopic Mises stress Macroscopic hydrostatic pressure
2. PAIRING the HALPIN‐TSAI eq. with the MODIFIED SECANT METHOD
PAIRING HALPIN‐TSAI WITH THE MODIFIED SECANT METHOD 1st
Which is equivalent to
1 1
1⁄ 1⁄
3
1⁄ 1⁄
• The Halpin‐Tsai equation provides explicit expressions for these derivatives in terms of 2nd
This scalar non‐linear equation is solved for , for any given macroscopic stress state (characterized by and ), using Newton‐Raphson.
TABLE OF CONTENTS 1. INTRODUCTION: CONTEXT and PREVIOUS WORK + Particulate Reinforced Metal Matrix Composites (PMMCs) + Summary of the MODIFIED SECANT METHOD for nonlinear composites. + A proposal: to use the HALPIN‐TSAI equation as the underlying required Linear Elastic Homogenization Scheme.
2. PAIRING the HALPIN‐TSAI equation with the MODIFIED SECANT METHOD + Halpin‐Tsai applied to Homogenization of elastic constants K and G. + Halpin‐Tsai applied to Homogenization of the secant constants K sec and G sec. + Resolution method.
3. APPLICATION EXAMPLE + Matrix phase with Ramberg‐Osgood hardening
3. APPLICATION EXAMPLES: matrix with experimentally measured hardening
MODELED COMPOSITE: MATRIX:
Ramberg‐Osgood hardening
Elastic properties:
Em = 71.1 GPa ; m = 0.34
Hardening parameters:
0 = 400 MPa ; 0 = 0.002 ; n = 10
Secant Shear Modulus:
PARTICLES: Linear Elastic Behavior Elastic properties:
Ep = 410 GPa ; p = 0.16
3. APPLICATION EXAMPLES: matrix with experimentally measured hardening
MODELED COMPOSITE: PARTICLE VOLUME FRACTION: F = 0 % (pure matrix) F = 10 % F = 20 % F = 30 %
MODELED LOAD CASE: MACROSCOPIC UNIAXIAL TENSION x
:
component of the macroscopic stress tensor in the loading direction (generated as a fictitious table, is user data)
q = x : macroscopic Mises stress. p = (1/3) x : macroscopic hydrostatic stress.
3. APPLICATION EXAMPLES: matrix with Ramberg‐Osgood hardening
RESULTS: 1. HARDENING of the COMPOSITE MATERIAL 700
Macro Uniaxial STRESS, MPa
600
500
400 Just MATRIX F = 0 % 300
Comp. F = 10 % Comp. F = 20 %
200
Comp. F = 30 % 100
0 0.00
0.01
0.02
0.03
Equivalent plastic strain
0.04
0.05
3. APPLICATION EXAMPLES: matrix with Ramberg‐Osgood hardening
RESULTS: 2. EVOLUTION of the HOMOGENIZED ‘SECANT’ PROPS.
90
40
80
35 30
SHEAR modulus, GPa
BULK modulus, GPa
70 60 50 40 30
Bulk modulus (composite)
20
Bulk modulus (matrix)
25 20 15 10
10
5
0
0
Shear modulus (composite) Shear modulus (matrix)
0
200
400
600
800
Macroscopic Tensile STRESS, MPa
0
200
400
600
800
Macroscopic Tensile STRESS, MPa
FINAL COMMENTS AND REMARKS
• It has been shown that the HALPIN‐TSAI linear elastic homogenization can be used in conjunction with the MODIFIED SECANT METHOD of Suquet (1996). • The approach was implemented under macroscopic STRESS CONTROL, but can also be implemented under macroscopic STRAIN CONTROL. • The approach does not use the Eshelby Tensor and approximate irregular and “edgy” particles by prismatic particles instead of the commonly used ellipsoids • Influence of “particle aspect ratio” is included by selecting appropriate values for the Halpin‐Tsai “S” parameter. • Using this approach, it was shown explicitly that HYDROSTATIC MACROSCOPIC LOADING has an influence on COMPOSITE’s plasticity. • COMPOSITE’s plastic strain DO HAVE a VOLUMETRIC component.
THANKS
Update: full paper available at http://dx.doi.org/10.1016/j.compstruct.2013.09.036
