Reversible multiscale homogenization of physical ...

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Reversible multiscale homogenization of physical properties of heterogeneous medium using GBSE 1

2

*Alexander Sokolov , Anton Pershin , Vitaly Schetinin1, Arseny Sapelkin1 1

2

Bauman Moscow State Technical University 1, Robotics and Complex Automation Department, 2-aya Baumanskaya, 5, 1, Moscow, Russia, E-mail: [email protected]

University of Leeds, Department of Applied Mathematics, Leeds, LS2 9JT, UK, E-mail: [email protected], [email protected]

CONTENTS

1. 2. 3.

4.

5.

6. 7.

ABSTRACT ABBREVIATIONS INTRODUCTION MAIN OBJECTIVES TASK STATEMENT 3.1. Estimation Of The Composite Material ElasticStrength Properties In Case Of Incomplete Input Data 3.2. Theoretical Conceptions 3.3. Procedure Of Effective Elastic Properties Determination 3.4. Procedure Of Strength Effective Properties Determination PROGRAM IMPLEMENTATION 4.1. GBSE Principles 4.2. Graph Model Of RMH Created Using GBSE 4.3. System Integration CALCULATION 5.1. Computational Task Statement Details 5.2. Computational Results DISCUSSION CONCLUSION

ABSTRACT The adaptive computational method (Reversible Multiscale Homogenization, RMH) for calculating the effective physical and mechanical properties of heterogeneous periodic medium (for example, composite materials) with an incomplete set of input data was proposed. New method is based on synchronous solving of direct and inversed tasks of micromechanics of composite materials using homogenization procedures and computational methods of multi-criteria optimization for solving inversed tasks. New method provides an automatic clarification of the numerical results with each successive calculation. Method was implemented using C/C++ based on special software framework named Graph-based Software Engineering Technology which was developed especially for this method and which was integrated into Distributed computational system GCD 2 (GCAD v.4). Computational results of elasticstrength properties determination using RMH for composite materials with different schemes of reinforcement were obtained. Keywords: material modeling, composite material, elasticity, strength, ultimate tensile strength, multiscale homogenization, elastic tensor, micromechanics, anysotropy, orthotropy, unit cell, finite element method, graph based software development. 1

URL: www.bmstu.ru (Bauman Moscow State Technical University). Since 1830. 2 Detailed information about research can be found at DCS GCD home page.

ABBREVIATIONS CM - Composite material CSG – Constructive Solid Geometry DCS GCD - Distributed Computational System Global Composite Development GBSE - Graph Based Software Engineering MH - Multiscale Homogenization Inv-MH – Procedure of MH In Inversed Formulation RMH - Reversible Multiscale Homogenization FEM – Finite Element Method PSO - Particle Swarm Optimization RVE – Representative Volume Element



INTRODUCTION

Composite materials (CM) are widely used in the mechanical engineering, construction industry, consumer industry, medical technologies (prosthetics, dentistry), shipbuilding and many other industries. For example, using carbon fiber-reinforced composite materials makes it possible to reduce overall weight of civil aircraft [13]. The flight duration can be increased with the same number of passengers in case of usage composite elements of aircraft construction. The new passenger aircraft Boeing 787 Dreamliner presents the example of efficient use of composite materials in aircraft engineering. This aircraft was designed and built of composites more than 50% of overall number of its construction parts. Using new heat-resistant composite materials makes it possible to increase speed of flight in the atmosphere at the same time. Composite materials are commonly used for heavily loaded parts (structural elements of wings (stringers), nozzle elements) of aircraft, boats, spacecraft and etc. (Figure 1).

Figure 1: Different applications of composite materials Carbon fiber reinforced plastics, carbon-carbon composites, glass-fiber composites, woven organoplastics based on complex monofilament fibers (Figure 2 (c)), N-D-reinforced composites (Figure 3 (a)), dispersedly-reinforced composites (Figure 3 (b)) and other new materials are used for all of these items. Pictures were taken by means of electron microscope FEI Phenom usage. One of the most important tasks which must be solved in the design process of new composite materials is the problem of estimation their physical properties. Most of the composite materials properties (strength, elastic, plastic, electrostatic and etc.) have been determined experimentally [1-3] or by using approximate formulae [1, 4-6]. Some researchers use highly specialized models [15, 16]. Full-scale investigations are quite expensive, while the

currently available formulae and estimations often fail to provide the required accuracy.

(a)

(b)

Unfortunately, often for real tasks part of the input D{α } is unknown. Moreover, some properties of composites can strongly depend on technological regimes in their production. For example, the elastic properties of a composite based on glass fibers and epoxy resin can be highly dependent on the curing regime of the epoxy resin.

D{1} ,, D{M } ,, D{ N }

(c)

Figure 2: (a) Fiber-glass composite with plain weave (scale level 1). (b) Single glass mono-fibers. Microstructure of composite with monofilament reinforcement structure (scale level 2). (c) Single carbon monofilament microstructure (scale level 3)

MH MH

GEO

D

cMlc

Figure 4: MH principal scheme, where D{α } - physical properties of component α of CM, GEO – geometric model describing the CM reinforcement scheme, D

calc

-

calculated averaged physical properties of CM



(a)

(b)

Figure 3: (a) Carbon-carbon 3D-orthogonal reinforced composite material microstructure. (b) Dispersedly reinforced composite material based on hollow microspheres The paper describes the fundamental research efforts aimed at the development of a new simulation method based on multiscale asymptotic homogenization techniques [7-10]. The practical value of proposed approach includes the following: 1) the opportunity to obtain results (effective properties of composites under investigation) with the required accuracy; 2) economic benefit through reduction of overall number of experimental investigations; 3) the opportunity to simulate the design process of new CM with desired properties; 4) improving the quality of industrial processes of production of new CM. One of the most widely used mathematical methods for analyzing the physical properties of composites is asymptotic averaging method or homogenization method also known as multiscale homogenization (MH) [7, 8, 10, 13, 14, 17]. Principal scheme of MH represented at Figure 4 including input and output data. The method MH provides an opportunity to obtain the average physical and mechanical properties of the composite structural element, which makes it possible to replace this element with a homogeneous one for further carrying out various engineering calculations of the entire composite structure. Such procedure provides ability to use standard finite element analysis significantly reducing the computational costs and in most cases makes it possible to analyze composite constructions at all. To obtain certain properties of the composite using MH it is demanded to know information about geometric model describing the composite reinforcement scheme (representative volume element (RVE)) as well as the respective properties of each component D{α } of the composite under research (Figure 4).

D = ( D{1} ,  , D{α } ,  , D{ N } )

T

(1)

MAIN OBJECTIVES

Main objective of research was to develop a new modification of homogenization method (MH) [7] to obtain physical properties of different types of composite materials with complex reinforcement multiscale inner structure in case of incomplete input set D{α } of initial data. This new method was named Reversible Multiscale Homogenization (RMH). New method RMH provides the possibility to analyze composite materials with complex multiscale structures (Figures 2, 3) in order to simulate physical processes in new advanced composite constructions. One of the requirements for new method was the need to take into account the specific features of the future technology of production of composite material while computational estimating its physical properties. In this paper for definiteness elastic-strength properties were considered for computational experiments. As a result of research, a new improved method RMH has been built. Due to its significant complexity special software technology was developed based on graph theory. This technology was named Graph Based Software Engineering (GBSE). Thus the software implementation of RMH was made using GBSE. Then it was integrated into Distributed Computational System GCD. A series of computational experiments has been conducted based on new method and computational results were presented.



TASK STATEMENT

As it was stated above, the objectives of the work were defined as follows: 1) to develop a mathematical model simulating the composite effective elastic and strength properties in terms of homogenization method (MH) theory [7]; 2) to build a corresponding computational method for estimating effective strength-elastic properties of composites (Young moduli, Shear moduli, Poisson's ratio, tensile strength limits strength limits

σc

στ

, compressive

and shear strength limits

ss

in

various planes and along various directions, ultimate limits and etc.); 3) to develop program algorithms of the new computational method RMH and their special software implementation;

4) to conduct a series of computational experiments with models of various types of models of composite materials (woven, fiber-reinforced, porous and dispersedly-reinforced composites were chosen) (Figure 2, 3, 5); 5) to obtain strength-elastic properties of composite materials under research with complex reinforcement multiscale inner structure in case when incomplete set of input data is defined; 6) to conduct a comparison analysis of computational results with test results; 7) to make a conclusion about the practicability of the developed computational procedures.

Estimation of effective properties of composite D

calc {α }

is

based on the known elastic and strength properties D of each of its components and use of the MH in case when full set of input data is provided. When we have insufficient data it was suggested to try to identify them beforehand using known experimental data. The main idea of new method is to incorporate experimental data into calculation scheme of MH procedure. It was made using MH procedure in inverse statement (Figure 4) in parallel with using MH in direct statement (Figure 5). Let define the incomplete input properties of components of CM in the form of vector

Dinc :

Dinc = ( D{1} ,  , D{α } ,  , D{incβ } ,  , D{incγ } ,  , D{ N } )

T

(2)

where

D{α } - fully specified input properties of component α ; (a)

D{incβ } , D{incγ }

(b)

β and

Figure 5: (a) RVE of woven composite with satin weave. (b) RVE of porous composite. Models were generated using CSG principles using automatic program tool integrated into DCS GCD

- incomplete input properties of components

γ.

{α }

incorporates elastic and strength data of All vectors D component α (see (1)).

Estimation Of The Composite Material Elastic-Strength Properties In Case Of Incomplete Input Data

γ} We will do identification of unknown D{incβ } , D{inc by means of solving inverse task of micromechanics of CM using procedure described at Figure 4.

Input data for MH procedure includes geometry data represents model of reinforcement scheme of CM (Figure 5) and mechanical properties of each component α of CM under research includes:

Procedure of solving inverse task have had to be previously implemented in most general case to be used for solving various identification tasks with another auxiliary composites.



1) elastic data:

Eiα

- Young Moduli;

Gijα

The procedure of solving single inverse task was designated as Inv-MH (see Figure 4).

- shear

α

moduli; ν ij - Poisson’s ratios;

The procedure of RMH represented at Figure 5 and includes in itself Inv-MH procedure.

2) special format was used to designate strength properties in the most general 3D case in 6-dimension space of stress tensor σ : σ xLMNPST , where

Composite under investigation was designated as



L, M , N , P, S , T ∈ {0, E , C} , E - extension, C -

compression, 0 – absence of load,

arbitrary auxiliary composite was designated as

x ∈ {e, u , y} ,

For RMH it was also assumed that

1)

It has been made as assumption that inner reinforcement structure of composite has periodic multiscale hierarchy type and it can be described by one or several unit cells or RVEs3.

{α }

α

CM 2

.

(auxiliary

must be produced using similar technology as CM 1

under investigation;

2) auxiliary composite can be found in commonly accessed database 4 and effective properties of this CM 2

T

Vector D component

CM 2

current

D{α } for all input data:

D{α } = ( Eiα , Gijα ,ν ijα , σαxLMNPST )

CM 2

must be exist

D{CM 2 ,tech}

{calc|exp}

, which

have had to be obtained earlier experimentally or by calculation;

(1)

3) auxiliary composite

represents all elastic-strength properties of of CM under investigation in short form.

CM 2

must include at least one of

the components β or γ of the CM 1 being investigated the properties of which are given in part (see (2)).

3

Scheme of reinforcement have to be defined as a set of RVE in multiscale case: each RVE for every scale level of the composite model and for every heterogeneous composite component.

and

composite) should meet the following requirements:

where e – elastic (proportional limit), u – ultimate, y – yield.

Let define special vector

CM 1

4

Relational database structure was implemented in DCS GCD.

In classical theory of multiscale homogenization method [7, 8, 9] special homogenization operator was introduced in form (3):

1 = ∫ f ( x, ξ)dv VRVE VRVE

f ( x, ξ ) =

where

x=x

coordinates,

L

(x)

f

GEO

DSβ } × DSγ }

(D

P2: P2: Select Select

Sβ } sel

- local normalized «fast» Cartesian

l

exp

P1: P1: Pre-Inv-MH Pre-Inv-MH

(3)

- global normalized «slow» Cartesian

ξ=x

D

Dinc

, DSselγ } ) ∈ DSβ } × DSγ }

coordinates, L – characterized dimension of CM area, l – characterized dimension of RVE area.



We used this operator

P3: P3: MH MH

to denote the properties of

Database

composites, obtained both numerically and experimentally for uniformity.

D

Let define

exp

D

and

calc

D{incβ } , D{incγ } to complete ones, where unknown parameters {β }

{γ }

are given in the form of interval estimates. D × D space of variable parameters induced by its interval estimates. D{selβ } , D{selγ } - an iteratively selected point from

)

the space of variable parameters complementing an incomplete set of input data

Dinc

up to the full

D . Pre-

Inv-MH – preprocessing procedure for inverse MH statement, produces

(D

{β } sel

, D{selγ } )

to use MH iteratively

(1) : ( DSselβ *} , DSselγ }* )

D

exp

and

exp

D

calc

(2) : ∅

Figure 4: Principal scheme of MH in inversed statement (Inv-MH), where data, D

exp

D{incβ }

and

D{incγ }

partially specified

- experimentally obtained properties of CM.

As a result of using Inv-MH unknown

D{incβ }

or

D{incγ } or

both of them can be identified using one or more auxiliary composites

CM 2

or may be

CM 3

. Thus the result can be

included into direct task, which can be solved using MH as it is represented at Figure 5.

D{incCM1 ,Pech}

Figure 4 shows the procedure for minimizing the between

∨ D

P4: P4: Compare Compare

(see Figure 4 for details).

discrepancy

cMlc

– effective properties of

CM obtained as a result of experiment and calculation respectively, GEO – geometry model of RVE of current CM, D{β } , D{γ } - addition of initial incomplete sets

(

D

(4)

GEO{CM1 }

tech

P1: P1: Pre-RMH Pre-RMH

(identification task of properties of components of CM – inverse MH task).

F

(D

exp

, D

calc

)=

D

exp

− D

calc

→ min

D{incCM 2 ,Pech}

(4)

To solve inverse MH task particle swarm optimization (PSO) [45] was used.

(D

{γ } sel*

,D

) was

identification task (inversed MH task). Each solution that was found using Inv-MH was stored in a common database of DCS GCD [46] for further use in the process of solving the whole problem of effective properties determination with incomplete input data set using RMH method.

It should also be noted that a specific task that must be solved using Inv-MH or RMH may not have a solution, which is also represented in the illustrations (Figure 4 (2), Figure 5 (2)).

GEO{CM 2 }

Database

D{selβ }

defined as a set of solutions of

Principal scheme of RMH as it was stated above represented at Figure 5.

{cMlc|exp}

P2: P2: Inv-MH Inv-MH

The task of identifying the properties of components of CM is an incorrectly posed (ill-posed) problem [49] and can have a multitude of solutions. In such a case the designation {β } sel*

D{CM 2 ,Pech}

P3: P3: MH MH

(1) : D{CM1 ,Pech}

cMlc

(2) : ∅

Figure 5: Principal scheme of RMH



Theoretical Conceptions

In this paper we discuss about elastic-strength properties of composites. In more detail, the model of the periodic medium and the concepts of the homogenization method are presented separately [38].



We consider a linear elasticity problem in displacements u in 3D case in vector formalization (5). The problem is stated in domain

VCM

with its external surface

Σ

expressed in

 terms of small strains theory, i.e. ∇ × u