Research ArticleNumerical Estimation of Effective Mechanical Properties forReinforced Plexiglas in the Two-Dimensional Case
Vladimir Levin,1 Ignatiy Vdovichenko,1 Anatoly Vershinin,1 Maksim Yakovlev,2
and Konstantin Zingerman3
1Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, GSP-1, 1 Leninskiye Gory, Main Building,Moscow 119991, Russia2Fidesys LLC, Office 402, 1 Building 77, MSU Science Park, Leninskie Gory, Moscow 119234, Russia3Tver State University, 33 Zhelyabov Street, Tver 170100, Russia
Correspondence should be addressed to Konstantin Zingerman; [email protected]
Received 1 May 2016; Accepted 25 July 2016
Academic Editor: Dimitrios E. Manolakos
Copyright Β© 2016 Vladimir Levin et al.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The paper describes an algorithm for numerical estimation of effective mechanical properties in two-dimensional case, consideringfinite strains.The algorithm is based on consecutive application of different boundary conditions to representative surface elements(RSEs) in terms of displacements, solution of elastic boundary value problem for each case, and averaging the stress field obtained.Effective properties are estimated as a quadratic dependence of the second Piola-Kirchhoff stress tensor upon the Green straintensor.The results of numerical estimation of effectivemechanical properties of plexiglas, reinforcedwith steel wire, are presented atfinite strains.Numerical calculationswere performedwith the help ofCAEFidesys using the finite elementmethod.Thedependenceof the effective properties of reinforced plexiglas upon the concentration of wires and the shape of wire cross section is investigated.In particular, it was found that the aspect ratio of reinforcing wire cross section has the most significant impact on effective modulicharacterizing the material properties in the direction of larger side of the cross section. The obtained results allow one to estimatethe influence of nonlinear effects upon the mechanical properties of the composite.
1. Introduction
Thewidespread use of plexiglas in engineering applications isdue to its properties such as transparency, strength, flexibility,lightness, cheapness, and nontoxicity. In some applicationsthe combination of plexiglas properties of high rigidity andresistance to mechanical and thermal effects is required. Inorder to increase stiffness and thermal conductivity of theplexiglas and to prevent its spillage due to mechanical orother influences a reinforcement wirework (usually made ofsteel) is used. Reinforced plexiglas is a composite material.Properties of the material obtained depend primarily uponthe material and shape of reinforcing wire. At that, thequestion that has to be answered is as follows: how toevaluate mechanical properties of reinforced plexiglas whilemechanical properties of plexiglas and wires are knownprovided the shape of wire cross section is known?
The averaging of heterogeneous materials propertieshas been of interest since the middle of the last century.Theoretical principles of such averaging are described in [1];in particular, a concept of representative volume element isexplained in detail. The studies of that time concerned theeffective properties of composite materials in the linear formsuitable for description of composites behavior under smallstrains.
For composites with relatively small volumetric contentof filler in the matrix, the existing bilateral assessment ofHashin and Shtrikman [2, 3] is valid; it gives the minimumand maximum values for the compression bulk modulusand a shear modulus of composite material (provided fillerconcentration and moduli of filler and matrix are known).Mori and Tanaka method [4] is an application of Hashinand Shtrikman conditions to fiber-reinforced material witha continuous matrix in which the volume fraction of the filler
Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2016, Article ID 9010576, 10 pageshttp://dx.doi.org/10.1155/2016/9010576
2 Modelling and Simulation in Engineering
particles is low, and the particles shape is mostly spherical.The book of Christensen [5] contains some analytical for-mulas for evaluating of effective elastic properties of fiber-reinforced, fibrous, and laminated composites and plasticand viscoelastic effects, and effective thermal properties ofcomposites are studied.
At present, the more topical problem is evaluation ofeffective mechanical properties of heterogeneous materialsin nonlinear form, with the help of which it is possible todescribe the behavior of composites under finite strains. Inthe work [6], elastic and plastic properties of the materialcontaining the dispersedmicrodefects of different orientationin space are studied. In articles [7, 8], effective elasticproperties of solids containing cavities of various shapesand orientations in space are estimated. In [9], for theconstruction of effective constitutive relations of viscoelasticmaterial with periodic structure the finite element methodis used; it serves to solve the two-dimensional problem ofelasticity theory for a representative volume element, andthen the results are averaged. Articles [10, 11] describe howto apply the variational principle to estimate the effectivecharacteristics of multicomponent composites in the form ofthe strain energy density. Work [12] presents a method ofeffective characteristics estimation of composite materials innonlinear form, based on the Tsukrov et al. [7, 8] principle.Work [13] describes a method for constructing nonlinearthermoviscoelastic effective constitutive relations for com-posites of a periodic structure with a rubber-like matrix.Work [14] describes the application of probability theorymethods for estimating of effective mechanical propertiesof composites having irregular structure. Work [15] presentsa method suitable for evaluating both elastic propertiesand thermal and electrical conductivity of fiber-reinforcedmaterials (the article compares the influence of differentparameters of reinforcing particles upon the effective elasticproperties and effective thermal and electrical conductivity).In [16], an averaging of elastic properties of inhomoge-neous material is performed under nonperiodic boundaryconditions considering geometric nonlinearity; and practi-cal implementation is carried out using the finite elementmethod (for two-dimensional case). In [17], this approach isapplied to the multiscale case. Article [18] compares differentmethods for properties averaging of both linear viscoelasticand nonlinear viscoplastic composite materials.
This work proposes an algorithm for construction ofeffective constitutive relations of composite materials innonlinear elastic form under finite strains. An evaluation ofthe effective characteristics is discussed for two-dimensionalcase, which is a simplification as compared to the three-dimensional case for which the algorithm is described in [19,20] in detail. Basic principles of this approach are as follows[21β23] (as for two-dimensional case). A rectangular RSE istaken. An effective material is a solid homogeneous materialwhich meets the following condition: if this homogeneousmaterial fills RSE and the source composite fills the exactsame RSE, then the average stress over the area in the originalcomposite and the effective homogeneous material are equalunder the identical displacements of boundaries of the RSE.
Constitutive relations for the effective material (i.e., the effec-tive properties) are plotted as a quadratic dependence of thesecond Piola-Kirchhoff stress tensor upon the Green straintensor. To calculate coefficients for this dependence, severalsequences of boundary value problems of the nonlinearelasticity should be solved for RSE with given displacementsof boundaries.
The paper presents the results of two-dimensional calcu-lations of reinforced plexiglas effective characteristics. Calcu-lations are performed with the help of CAE Fidesys using thefinite element method. Plexiglas and its reinforcing steel wirewere modeled by the Murnaghan material. A dependenceof the effective elastic moduli on the concentration of thereinforcing wire and the shape of its cross section wasinvestigated.
2. Materials and Methods
2.1. Algorithm of Numerical Evaluation of EffectiveMechanicalProperties of Composite in Two-Dimensional Case at FiniteStrains. Let us present the basic designations and relationsof the nonlinear theory of elasticity [24] which will beused in describing the algorithm of numerical evaluation ofeffective mechanical properties of composite material in two-dimensional case at finite strains:
0
π , π : radius vector of a particle in initial and currentstates;
π’ = π β0
π : displacement vector;ππ: material coordinates of a particle;π₯π: spatial coordinates of a particle;ππ: basis vectors of reference system;0
qπ= π
0R/πππ, q
π= ππ /ππ
π: basis vectors in initial andcurrent states;0
β =0
qππ/πππ, β = q
ππ/πππ: gradient operators in initial
and current states;πΌ: identity tensor;
Ξ¨ =0βπ = πΌ +
0βπ’ = (βπ)
β1= (πΌ β βπ’)
β1: deformationgradient;0
πΈ = (1/2)(Ξ¨ β Ξ¨β β πΌ) = (1/2)(0βπ’ + π’
0β+
0βπ’ β π’
0β): the
Green strain tensor;πΈ = (1/2)(πΌ βΞ¨
β1β Ξ¨ββ1) = (1/2)(βπ’+π’βββπ’ β π’β):
the Almansi strain tensor;Ξ = detΞ¨ β 1: relative change of volume;β: sign of transposition;π: true stress tensor;R = (1 + Ξ)Ξ¨
ββ1β π: the first Piola-Kirchhoff stress
tensor;0
β = (1 + Ξ)Ξ¨ββ1 β π β Ξ¨β1: the second Piola-Kirchhoffstress tensor;π0, π: area of RSE in initial and current states;
Modelling and Simulation in Engineering 3
Ξ0, Ξ: boundary of the RSE in initial and current states;0
π, π: normal to the boundary in initial and currentstates.
Let us call the effective (averaged)material such a homoge-neous material that will meet the following condition: if thishomogeneous material fills RSE and the source compositefills the exact same RSE, then average (over the area) stressesin the original composite and in the effective homogeneousmaterial are equal under identical displacement of faces. Letus call mechanical properties of this material as effectiveproperties.
Using the above definitions and designation, let usdescribe the algorithm for estimating effective characteristicof composite material for two-dimensional case in the non-linear form under finite strains.
For RSE π0in the initial state (before deformation), we
will solve a certain number of sequences of boundary valueproblems of nonlinear theory of elasticity [24]:
β β π = 0
or0
β β R = 0
(1)
with boundary conditions
π’|Ξ0= π β (Ξ¨
πβ πΌ) . (2)
Each sequence of problems corresponds to a certain typeof boundary conditions (2) and a certain type of effectivestrain tensorπΈπ over the RSE (i.e., to a certain type of effectivedeformation gradient Ξ¨π). Different problems within thesame sequence differ by strain values.
Solving each problem of each sequence we will find thefield of the true stress tensor π. With the knowledge of it wewill calculate the effective stress tensor ππ by averaging of πover the area:
ππ=1
πβ«π
πππ =1
πβ«Ξ
π β ππ πΞ. (3)
The second equality in (3) was obtained with the helpof divergence theorem and the fact that the RSE is inequilibrium:
β β (ππ ) = (β β π) π + π (β β π )β= (β β π) π + π β πΌ
= π.
(4)
Knowing the deformation gradient that was specified in(2), it is possible to calculate the effective Green strain tensorusing the following formula:
0
πΈπ=1
2(Ξ¨πβ Ξ¨πββ πΌ) . (5)
In a linear case, the effective constitutive relations areestimated as a linear dependence of true stress tensor ππ uponstrain tensor:
ππ
ππ= πΆπππππΈπ
ππ. (6)
In the nonlinear case, the obtained effective true stresstensor ππ is used first for calculation of the effective secondPiola-Kirchhoff stress tensor
0
Ξ£π:
0
Ξ£π= (detΞ¨π) (Ξ¨π)ββ1 β ππ β (Ξ¨π)β1 . (7)
In this case, the effective properties are estimated asquadratic dependence of the second Piola-Kirchhoff stress
tensor0
Ξ£π upon the Green strain tensor πΈπ:
0
Ξ£π
ππ=(0)
πΆ ππππ
0
πΈπ
ππ+
(1)
πΆππππππ
0
πΈπ
ππ
0
πΈπ
ππ. (8)
Thus, estimation of the effective properties of the compos-ite in linear case reduces to the calculation of coefficientsπΆ
ππππ
(6), and estimation of effective properties in nonlinear case
reduces to calculation of coefficients(0)
πΆ ππππ and(1)
πΆππππππ
(8).At that, the following conditions of symmetry are in force
for tensor components πΆππππ
in (6):
(1) πΆππππ= πΆππππ
(due to symmetry of stress tensor);(2) πΆππππ= πΆππππ
(due to symmetry of strain tensor);(3) πΆππππ
= πΆππππ
(due to existence of strain energydensity).
The same conditions are in force for(0)
πΆ ππππ in (8). For(1)
πΆππππππ
in (8) the following conditions are in force:
(1)(1)
πΆππππππ
=(1)
πΆππππππ
(due to symmetry of strain tensor);
(2)(1)
πΆππππππ
=(1)
πΆππππππ
(due to symmetry of strain tensor);
(3)(1)
πΆππππππ
=(1)
πΆππππππ
(due to symmetry of multiplication);
(4)(1)
πΆππππππ
=(1)
πΆππππππ
(due to symmetry of the Piola-Kirchhoff tensor).
Considering the above conditions of symmetry, there are
21 independent constants πΆππππ
or(0)
πΆ ππππ and 126 independent
constants(1)
πΆππππππ
.
2.2. Finite Element Implementation of the Algorithm. Imple-mentation of the algorithm relies on the use of finite elementmethod [25, 26] for the calculations. For numerical estima-tion of effective properties, a geometric model of RSE of acomposite material is required, which should be βcut outβ inthe shape of a rectangle whose edges should be parallel to thecoordinate planes.The finite element mesh, which is an inputfor the algorithm, is constructed on this geometric model.Further actions are performed in the Cartesian coordinatesystem, coordinate planes of which are parallel to the edgeof RSE. In this coordinate system, the effective mechanicalcharacteristics will be evaluated.
4 Modelling and Simulation in Engineering
The algorithm consists of the following logical blocks.
(1) Preparing the Mesh for Calculation. The finite elementmesh includes an array of node numbers with their coordi-nates and an array of numbers of finite elements with thenumbers of their constituent nodes. In order to determinethe overall dimensions of the RSE, a calculation loop over allnodes of the finite element mesh is conducted, which definesthemaximum andminimum abscissas and ordinates of meshnodes: πmax, πmin, πmax, and πmin. Then the coordinates ofthe RSE center are calculated:
πcenter =1
2(πmax + πmin) ,
πcenter =1
2(πmax + πmin) .
(9)
After that, the model is shifted to put its center to theorigin (i.e., the coordinates of all mesh nodes are decreasedby πcenter and πcenter). Also, overall dimensions of the model2π΄ Γ 2π΅ are determined by formulas:
π΄ =1
2(πmax β πmin) ,
π΅ =1
2(πmax β πmin) .
(10)
In addition, the accuracy π for determination of boundarynodes and edges is calculated for the same block. For thispurpose a calculation loop over all finite elements of themeshis conducted, which defines the minimum area of an elementπmin. The accuracy π is calculated by formula
π =
3βπmin100.0
. (11)
The value of π naturally depends on how finemesh is usedfor calculation.
(2) Formation of Lists of Boundary Nodes. For further calcu-lations it is necessary to form a list of nodes on boundariesof the RSE. Particular lists depend on type of boundaryconditions applied.
If the problem is solved for nonperiodic boundary condi-tions, all nodes located on the boundaries are put to commonlist. For this purpose the calculation loop is conducted overall mesh nodes, at each step of which the following conditionis checked: |π₯ β π΄| < π OR |π₯ + π΄| < π OR |π¦ β π΅| < π OR|π¦ + π΅| < π. Here, π₯ and π¦ are coordinates of the node. If thecondition is true, this means that the node is on the boundaryof the RSE, and its number is added to the list of boundarynodes.
If the problem is solved for periodic boundary conditions,four lists of numbers of nodes are created (according tonumber of edges). Similarly, the calculation loop is conductedover all mesh nodes, at each step of which the followingcondition is checked:
(1) |π₯ β π΄| < π;(2) |π₯ + π΄| < π;
(3) |π¦ β π΅| < π;(4) |π¦ + π΅| < π.
If the conditionwith number π is true for a node, the nodenumber is added to the list with the number π.
Then, a pair of corresponding nodes (i.e., those, theprojection of which is closest to each other) are formed fromopposite edges of the RSE.The pairs are recorded to two lists.
For this purpose, a pass through all nodes of the edge π₯ =π΄ is conducted. For each such node π a node βπ is found: theclosest one to the point on the edge π₯ = βπ΄ that is oppositeto the node π. In other words, for the node π with coordinates(π΄, π¦π) (which lies on the edge π₯ = π΄) the closest one (from
nodes lying on the edgeπ₯ = βπ΄) to the pointwith coordinates(βπ΄, π¦
π) is searched for. When such a node is found, the pair
(π, βπ) is added to the first list of pairs of nodes.Then the loopis conducted over all nodes of the edge π₯ = βπ΄: if the node βπof this edge is already the second element of one of the pairs,we can proceed to the next one; but if not, then you need tofind its corresponding node π on the edge π₯ = π and add it tothe first list of pairs.
Similarly, a pass through all nodes of the edge π¦ = π΅
is conducted. For each such node π a node βπ is found: theclosest one to the point on the edge π¦ = βπ΅ that is oppositeto the node π. In other words, for the node πwith coordinates(π₯π, π΅) (which lies on the edge π¦ = π΅) the closest one (from
nodes lying on the edgeπ¦ = βπ΅) to the point with coordinates(π₯π, βπ΅) is searched for. When such a node is found, the pair
(π, βπ) is added to the second list of pairs of nodes. Then theloop is conducted over all nodes of the edge π¦ = βπ΅: if thenode βπ of this edge is already the second element of one ofthe pairs, we can proceed to the next one; but if not, then youneed to find its corresponding node π on the edge π¦ = π΅ andadd it to the second list of pairs.
(3) Formation of Lists of Boundary Edges. For furtheraveraging over an edge it is necessary to form a listof edges of finite elements on the boundaries of theRSE. Since the finite element mesh data do not usu-ally contain an array of edges of elements, four lists ofpairs {global element number; local number of the edges}are formed. For this purpose a calculation loop over all finiteelements of themesh is conducted, and a loop over all edges ofthe element inside it is conducted, too. If all nodes of (local)element edges lie on one of the edges of the RSE (global), apair {global element number; local number of the edges} isadded to an appropriate list. This takes into account that theelements are of different geometric shape and different orderof approximation; that is, different elements have differentnumber of edges and different number of nodes on them.
(4) Application of Boundary Conditions. For RSE we willsolve a certain six sequences of boundary value problemsof nonlinear theory of elasticity [24]. Different problemsequences differ by types of applied boundary conditions(i.e., type of effective strain tensor over the RSE). Differentproblems within the same sequence differ in the amount ofstrain while being of the same type. The following types ofstrains are applied:
Modelling and Simulation in Engineering 5
(1)0
πΈ11 = π: strain or compression along the axisπ;
(2)0
πΈ22 = π: strain or compression along the axis π;
(3)0
πΈ12 =0
πΈ21 = π: shear inππ plane;
(4)0
πΈ11 =0
πΈ22 = π: composition of tensions orcompressions along two axes:π and π;
(5)0
πΈ11 = π;0
πΈ12 =0
πΈ21 = π: composition of tensions orcompressions along the axisπ and shear inππ plane;
(6)0
πΈ22 = π;0
πΈ12 =0
πΈ21 = π: composition of tensions orcompressions along the axis π and shear inππ plane,
where π is the amount of strain.There are 6 sequences, each of which contains 3-4
problems or more, if required. Different problems within thesame sequence differ in the amount of strain π.
On the basis of effective strain tensor, the effectivedeformation gradient Ξ¨ is calculated by formula (5). Sincethe strain tensor is symmetric and deformation gradient incommon case is asymmetric, for certainty it is assumed thatthe gradient is upper triangular. Then (5) can be figured outby components in the following form:
(
π11π12
π12π22
)
=1
2[(
π11π12
0 π22
)(
π11
0
π12π22
) β (
1 0
0 1)] .
(12)
Formulas for gradientβs components in an explicit formare as follows:
π22= β2π
22+ 1,
π12=2π22
π22
,
π11= β2π
11+ 1 β π
2
12.
(13)
For nonperiodic boundary conditions there is one com-mon list of numbers of all nodes on the boundary of the RSE.Boundary condition (2) is applied to each node in this list inthe form of rigidly given displacements in all two directions.
For periodic boundary conditions there are two listsof pairs of nodes on the boundary of the RSE. For eachpair of nodes from the first list (π, βπ) the relationship withdisplacement is specified in the following form:
π’πβ π’βπ= β2π΄ (π
11β 1) ,
Vπβ Vβπ= β2π΄π
12.
(14)
For each pair of nodes from the second list (π, βπ) therelationship with displacement is specified in the followingform:
π’πβ π’βπ= β2π΅π
21,
Vπβ Vβπ= β2π΅ (π
22β 1) .
(15)
(5) Solution of Boundary Value Problem of Elasticity Theoryand Averaging of Results. After application of boundaryconditions to the RSE, the actual numerical solution of eachboundary value problem of the elasticity theory of eachsequence is performed using the finite element method.When the solution is found, it is necessary to average obtainedfield of stress tensor over the area.
First, the area of the model in the final state (i.e., afterdeformation) is calculated using the formula:
π = 4π΄π΅ β detΞ¨. (16)
In other words, the area of model in final state representsthe area in initial state multiplied by the determinant ofeffective deformation gradient.
Stress tensor is averaged over the area using formula(3). Since the integration is performed on the finite elementmodel, a transition from integral over the entire boundary tothe sum of the integrals is done over all boundary edges:
ππ=1
πβ«π
πππ =1
πβ«Ξ
π β ππ πΞ
=1
πβ
π
β«πΎπ
π β ππ ππΎπ.
(17)
Hereby one can calculate the effective true stresses tensorin the RSE. Knowing it, it is possible to calculate the effectivesecond Piola-Kirchhoff stress tensor using formula (7).
Blocks 4 and 5 of the algorithm should be run in doubleloop: by type of strain (1 to 6) and by amount of strain(from 1 to 3. . .4 or more). The result is as follows: for eachboundary value problem for each sequence an effectiveGreenstrain tensor was set, and as a result the effective secondPiola-Kirchhoff stress tensor was obtained. The resultingPiola-Kirchhoff tensors are stored for each problem forlater calculation of effective properties in the form of theirrelations.
(6) Plotting of Piola-Kirchhoff Tensor versus Green TensorUsing the Least Squares Method. Since the problems withinone sequence differ by the amount of strain only, for eachproblem of each sequence the dependence of the effectivesecond stress Piola-Kirchhoff tensor versus characteristicamount of strain π is plotted:
0
Ξ£π
ππ= πΌ0
πππ + πΌ1
πππ2. (18)
The dependence is constructed using the least squaresmethod, which allows calculating coefficients πΌ0
ππand πΌ1
ππ.
(7) Calculation of Effective Elastic Moduli of First and SecondOrders. In block 6 for each problem of each sequence, we haveconstructed the dependence (18) of the effective secondPiola-Kirchhoff stress tensor versus the characteristic amount ofstrain, π. Effective properties are evaluated as (8).
The relationship between the calculated coefficients πΌ0ππ
andπΌ1ππin (18) for each problem sequence anddesired effective
elastic moduli πΆ0ππππ
(of the first order) and πΆ1ππππππ
(of thesecond order) is as follows:
6 Modelling and Simulation in Engineering
(1)0
πΈπ = ( π 00 0) β
0
Ξ£π
ππ= πΆ0ππ11π + πΆ1ππ1111
π2 =(1)
πΌ0
πππ +(1)
πΌ1
πππ2;
(2)0
πΈπ = ( 0 00 π ) β
0
Ξ£π
ππ= πΆ0ππ22π + πΆ1ππ2222
π2 =(2)
πΌ0
πππ +(2)
πΌ1
πππ2;
(3)0
πΈπ = ( 0 ππ 0) β
0
Ξ£π
ππ= (πΆ0ππ12+ πΆ0
ππ21)π + 4πΆ1
ππ12π2 =(3)
πΌ0
πππ
+(3)
πΌ1
πππ2;
(4)0
πΈπ = ( π 00 π) β
0
Ξ£π
ππ= (πΆ0ππ11+πΆ0
ππ22)π+(πΆ
1
ππ1111+πΆ1
ππ2222+
πΆ1
ππ1122+ πΆ1
ππ2211)π2 =(4)
πΌ0
πππ +(4)
πΌ1
πππ2;
(5)0
πΈπ = ( π ππ 0 ) β
0
Ξ£π
ππ= (πΆ0ππ11
+ 2πΆ0
ππ12)π + (πΆ1
ππ1111+
4πΆ1
ππ1112+ 4πΆ1
ππ1212)π2 =(5)
πΌ0
πππ +(5)
πΌ1
πππ2;
(6)0
πΈπ = ( 0 ππ π ) β
0
Ξ£π
ππ= (πΆ0ππ22
+ 2πΆ0
ππ12)π + (4πΆ1
ππ1212+
4πΆ1
ππ1222+ πΆ1
ππ2222)π2 =(6)
πΌ0
πππ +(6)
πΌ1
πππ2.
Formulas for calculation of coefficientsπΆ0ππππ
in an explicitform are as follows:
(1) πΆππ11
=(1)
πΌ0
ππ;
(2) πΆππ22
=(2)
πΌ0
ππ;
(3) πΆππ12= πΆππ21
=(3)
πΌ0
ππ/2.
For calculation of coefficients πΆ1ππππππ
it is necessary tosolve the system of 6 linear algebraic equations:
(1) πΆ1ππ1111
=(1)
πΌ1
ππ;
(2) πΆ1ππ2222
=(2)
πΌ1
ππ;
(3) 4πΆ1ππ1212
=(3)
πΌ1
ππ;
(4) πΆ1ππ1111
+ 2πΆ1
ππ1122+ πΆ1
ππ2222=(4)
πΌ1
ππ;
(5) πΆ1ππ1111
+ 4πΆ1
ππ1212+ 4πΆ1
ππ1112=(5)
πΌ1
ππ;
(6) πΆ1ππ2222
+ 4πΆ1
ππ1212+ 4πΆ1
ππ1222=(6)
πΌ1
ππ.
This system is solved analytically, and the solution is asfollows:
πΆ1
ππ1111=(1)
πΌ1
ππ; πΆ1ππ2222
=(2)
πΌ1
ππ; πΆ1ππ1212
= (1/2)(3)
πΌ1
ππ;
πΆ1
ππ1122= (1/2)(
(4)
πΌ1
ππβ
(2)
πΌ1
ππβ
(1)
πΌ1
ππ);
πΆ1
ππ1112= (1/2)(
(5)
πΌ1
ππβ
(3)
πΌ1
ππβ
(1)
πΌ1
ππ);
πΆ1
ππ1222= (1/2)(
(6)
πΌ1
ππβ
(3)
πΌ1
ππβ
(2)
πΌ1
ππ).
Thus, the described algorithm allows performing numer-ical (using the finite element analysis) estimation of effectiveelastic moduli (of the first and the second order) for thecompositematerial in the two-dimensional case. On the basisof the described algorithm we have developed a softwaremodule Fidesys Composite as part of CAE system Fidesys[27] intended for numerical estimation of effective mechani-cal properties of composite materials.
The proposed algorithm assumes that the computationsfor each geometrical pattern are performed separately. Forcomposites of periodical structure it is sufficient to performcomputations once. For composites of random structure withstatistically uniformdistribution of inclusions one can use theapproach that is based on the ensemble averaging over a fixednumber of configurations [21, 22]. The additional averagingover all possible directions in the plane of deformation [21,22] can be performed analytically and results in effectiveconstitutive equations for transversely isotropic materials.These methods of averaging permit one to avoid the hugeamount of computations for irregular composites.
3. Results
With the help of the developed module Fidesys Composite,two series of finite element analyses of effective characteristicsof plexiglas reinforced with steel wire were conducted fortwo-dimensional case in the nonlinear form under finitestrains. A rectangular RSE of plexiglas with a rectangle of steelmodeling wire was considered in all calculations. A level ofstrains applied to the RSE was 1%, 2%, and 3%. Mechanicalproperties of plexiglas and steel were described using theMurnaghan constitutive relations [24]:
0
Ξ£ = π(
0E : πΌ) + 2πΊ
0E + 3πΆ
3(
0E : πΌ)
2
πΌ + πΆ4(
0E2 : πΌ) πΌ
+ 2πΆ4(
0E : πΌ)
0E + 3πΆ
5
0E2.
(19)
The constants π = 1.09 β 105MPa, πΊ = 0.818 β 105MPa,πΆ3= β0.29 β 105MPa, πΆ
4= β2,4 β 105MPa, and πΆ
5= β2.25
β 105MPa [24] were used for steel, and the constants π = 0,39β 105MPa, πΊ = 0.186 β 105MPa, πΆ
3= β0.013 β 105MPa, πΆ
4=
β0.07 β 105MPa, and πΆ5= 0.063 β 105MPa [24] were used for
plexiglas.
3.1. Dependence of Effective Properties versus Concentrationof Wires. Dependence of effective properties of reinforcedplexiglas versus the concentration of the reinforcing wireswas studied. RSE was a rectangle 10 Γ 5mm. Cross sectionof wire is square; size of square side ranged from 0.25mm to3mm.
Modelling and Simulation in Engineering 7
74000760007800080000820008400086000880009000092000
0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00Wire concentration (%)
C1111
(MPa
)
Figure 1: Dependence of coefficient(0)
πΆ1111 for reinforced plexiglasversus concentration of reinforcing wires.
38500390003950040000405004100041500420004250043000
0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00Wire concentration (%)
C1122
(MPa
)
Figure 2: Dependence of coefficient(0)
πΆ1122 for reinforced plexiglasversus concentration of reinforcing wires.
Let us see graphs for linear coefficients(0)
πΆ1111,(0)
πΆ1122,(0)
πΆ1212
and(0)
πΆ2222 (see Figures 1β4).
The graphs show that the coefficient(0)
πΆ1111 definingthe material behavior during the deformation along theaxis π almost linearly depends on concentration of wires
(Figure 1): the greater the concentration, the more(0)
πΆ1111. The
dependence of coefficient(0)
πΆ2222 responsible for the behaviorof the material during the tension along the axis π versusthe concentration is similar (Figure 4). Similarly, coefficients(0)
πΆ1212 (Figure 3) defining the behavior of the material under
shear strain and coefficient(0)
πΆ1122 (Figure 2) are dependentupon concentration of wires.
Let us see graphs for nonlinear coefficients(1)
πΆ111111,(1)
πΆ111122, and(1)
πΆ222222 as well (see Figures 5β7).
As can be seen from the graphs, the coefficient(1)
πΆ111111
(which determines the nonlinearity of the material duringtension along the axis π) increases in modulus with increas-ing of concentration of wires (Figure 5). The coefficient(1)
πΆ111111 (responsible for the nonlinearity during tension alongthe axis π) increases in modulus even stronger, and thedependence of this coefficient on concentration of wires isclearly nonlinear (Figure 7): this is probably due to the factthat the width of the RSE is half as much as length. Other
15000160001700018000190002000021000220002300024000
0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00Wire concentration (%)
C1212
(MPa
)
Figure 3: Dependence of coefficient(0)
πΆ1212 for reinforced plexiglasversus concentration of reinforcing wires.
6000065000700007500080000850009000095000
100000
0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00Wire concentration (%)
C2222
(MPa
)
Figure 4: Dependence of coefficient(0)
πΆ2222 for reinforced plexiglasversus concentration of reinforcing wires.
β16000β14000β12000β10000β8000β6000β4000β2000
0
0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00Wire concentration (%)
C111111
(MPa
)
Figure 5: Dependence of coefficient(1)
πΆ111111 for reinforced plexiglasversus concentration of reinforcing wires.
β12000
β10000
β8000
β6000
β4000
β2000
0
0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00Wire concentration (%)
C111122
(MPa
)
Figure 6: Dependence of coefficient(1)
πΆ111122 for reinforced plexiglasversus concentration of reinforcing wires.
8 Modelling and Simulation in Engineering
β40000β35000β30000β25000β20000β15000β10000β5000
0
0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00Wire concentration (%)
C222222
(MPa
)
Figure 7: Dependence of coefficient(1)
πΆ222222 for reinforced plexiglasversus concentration of reinforcing wires.
75000
80000
85000
90000
95000
100000
105000
1/1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10Width/length of inclusion
C1111
(MPa
)
Figure 8: Dependence of coefficient(0)
πΆ1111 for reinforced plexiglasversus form of reinforcing wires cross section.
nonlinear coefficients (e.g.,(1)
πΆ111122, the graph for which isshown in Figure 6) depend on concentration significantlylesser.
3.2. Dependence of Effective Properties versus the Form of CrossSection. Dependence of effective properties of reinforcedplexiglas versus the form of cross section of the reinforcingwires was studied. RSE was a square 10 Γ 10mm. Therectangular form of wire cross section was considered, theratio of rectangle sides varied from 1 : 1 to 1 : 10. At that, thecross-section area of the wire was 9mm2.
Let us see graphs for linear coefficients(0)
πΆ1111 and(0)
πΆ2222
(see Figures 8 and 9).
The graphs show that the coefficient(0)
πΆ1111 grows mono-tonically and almost linearly as the cross section of thereinforcing wire is βpulledβ along the axis π (Figure 8). At
that, the coefficient(0)
πΆ2222 decreases (Figure 9) monotonicallybut nonlinearly: at first, this coefficient falls sharply (with thechange of aspect ratio from 1 : 1 to 1 : 3) and then changes
slightly. Other linear coefficients(0)
πΆ ππππ are slightly dependentupon the shape of wires section as shown by calculations.
Let us see graphs for nonlinear coefficients(1)
πΆ111111,(1)
πΆ121212, and(1)
πΆ222222 (see Figures 10β12).
80000805008100081500820008250083000835008400084500
1/1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10Width/length of inclusion
C2222
(MPa
)
Figure 9: Dependence of coefficient(0)
πΆ2222 for reinforced plexiglasversus form of reinforcing wires cross section.
β180000β160000β140000β120000β100000β80000β60000β40000β20000
0
1/1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10Width/length of inclusion
C111111
(MPa
)
Figure 10:Dependence of coefficient(1)
πΆ111111 for reinforced plexiglasversus form of reinforcing wires cross section.
β400β350β300β250β200β150β100β50
0
1/1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10Width/length of inclusion
C121212
(MPa
)
Figure 11: Dependence of coefficient(1)
πΆ121212 for reinforced plexiglasversus form of reinforcing wires cross section.
β12000
β10000
β8000
β6000
β4000
β2000
0
1/1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10Width/length of inclusion
C222222
(MPa
)
Figure 12: Dependence of coefficient(1)
πΆ222222 for reinforced plexiglasversus form of reinforcing wires cross section.
Modelling and Simulation in Engineering 9
Table 1: Linear effective elastic modulus of the fiber-reinforcedcomposite. A comparison between the Fidesys computations and theresults of Christensen [5].
Effective moduli, MPa πΆ1111
πΆ1122
πΆ1212
πΆ2222
Fidesys 3.13691 1.30607 0.89212 3.13692Christensen [5] 3.11029 1.33286 0.88872 3.11029
As can be seen from the graphs, the coefficient(1)
πΆ111111
increases in modulus monotonically as you βpullβ the crosssection of the wires along the axis π, (Figure 10), and thedependence is close to linear one. At that, the module of
coefficient(1)
πΆ222222 decreases (Figure 12) also monotonically
but nonlinearly: similar to the coefficient(0)
πΆ2222, the coeffi-
cient(1)
πΆ222222 decreases in modulus strongly with decreasingof aspect ratio of the rectangle from 1 : 1 to 1 : 3; then it changes
weakly. The graph for the coefficient(1)
πΆ121212, defining thenonlinearity of material under shear strains, looks morecomplicated (Figure 11): first, upon the increase of the aspectratio of the rectangular cross section, the modulus of thiscoefficient drops by approx. six times, and then it rises againby seven times. The minimum of modulus is observed whenthe aspect ratio is approximately 1 : 6.
The points of inflection in the figures showing stiffnessconstants as a function of the width/length ratio of theinclusions arise apparently due to computational errors.
3.3. Comparison with Results Obtained by Other Methods.Within the framework of linear elasticity and small strains,the obtained results are compared with the results of Chris-tensen [5]. The materials of matrix and fibers (wires) wereassumed to be isotropic, and the comparison was performedfor the plane strain. In our computations RSE was a square,and the circular inclusion was in the center of RSE. The con-centration of the reinforcing wire was 10%. The mechanicalproperties of inclusions (wires) are Youngβs modulus πΈ =2000MPa and Poissonβs ratio ] = 0.2.Themechanical proper-ties of a matrix are Youngβs modulus πΈ = 2MPa and Poissonβsratio ] = 0.3. The effective elastic modulus is given in Table 1.
One can see from Table 1 that the difference between thenumerical solution and the analytical results given in [5] doesnot exceed 3%.
Another comparison was performed for the case of finitestrains. The pure shear of a fiber-reinforced elastomericcomposite with rigid circular fibers (wires) was considered.The matrix material was incompressible, and the mechanicalproperties of the matrix material were described by the neo-Hookean potential. The stresses were averaged over the RSE,and the first Piola shear stress was computed. The resultswere comparedwith the estimations of Avazmohammadi andPonte Castaneda [28]. These results are shown in Table 2.In this table Ξ is a shear strain and π is the concentrationof wires. The results of the comparison may be estimated assatisfactory.
Table 2: The first Piola-Kirchhoff shear stress in a fibrous elas-tomeric composite as a function of shear strain. A comparisonbetween the Fidesys computations and the results of Avazmoham-madi and Ponte Castaneda [28].
Ξ
π = 0.1 π = 0.2
FidesysAvazmohammadi
and PonteCastaneda
FidesysAvazmohammadi
and PonteCastaneda
1.25 0.83 0.94 1.34 1.121.50 1.93 1.48 2.81 1.92
4. Discussion
Thus, the paper presents the algorithm for numerical eval-uation of the effective mechanical properties of nonlinearelastic solids in two-dimensional case (under plane strain).The novelty of this algorithm is determined by considering ofnonlinear effects. Both physical and geometric nonlinearityis taken into account. Effective constitutive relations arepresented in the form of quadratic dependence of averagedstrains versus stresses. Determination of effective modulesis reduced to solution of sequences of nonlinear boundaryvalue problems of elasticity for loads of different types andsizes. For solving boundary value problems the finite elementmethod is used, it is implemented in CAE system Fidesys.Results of calculation of reinforced plexiglas effective char-acteristics, presented in the paper, confirm the efficiency ofthe algorithm. The influence of concentration of reinforcingwires and wire shape upon the effective properties is studied.In particular, it was found that the aspect ratio of reinforcingwire cross section has themost significant impact on effectivemoduli characterizing thematerial properties in the directionof larger side of the cross section.
The obtained results allow estimating the influence ofnonlinear effects upon the mechanical properties of thecomposite for different amounts of strain. For example, inFigures 8 and 10 one can see that for elongation of 10% inthe direction of the axis π₯
1the amendment for adoption of
nonlinear effects for the stress in the direction of the sameaxis is 1% for rectangular wires, and for wires cross sectionwith an aspect ratio of 10 : 1 this amendment will amount toca. 16% under the same strain.
In the future it is planned to perform similar calculationsfor other composite materials.
Competing Interests
The authors declare no competing interests.
Acknowledgments
The research for this paper was performed in Fidesys LLCand was financially supported by Russian Ministry of Edu-cation and Science (Project no. 14.579.21.0076; Project IDRFMEFI57914X0076).
10 Modelling and Simulation in Engineering
References
[1] R. Hill, βElastic properties of reinforced solids: some theoreticalprinciples,β Journal of theMechanics and Physics of Solids, vol. 11,no. 5, pp. 357β372, 1963.
[2] Z. Hashin and S. Shtrikman, βOn some variational principlesin anisotropic and nonhomogeneous elasticity,β Journal of theMechanics and Physics of Solids, vol. 10, no. 4, pp. 335β342, 1962.
[3] Z. Hashin and S. Shtrikman, βA variational approach to thetheory of the elastic behaviour of multiphase materials,β Journalof the Mechanics and Physics of Solids, vol. 11, no. 2, pp. 127β140,1963.
[4] T. Mori and K. Tanaka, βAverage stress in matrix and averageelastic energy of materials with misfitting inclusions,β ActaMetallurgica, vol. 21, no. 5, pp. 571β574, 1973.
[5] R. M. Christensen, Mechanics of Composite Materials, Wiley-Interscience, New York, NY, USA, 1979.
[6] O. T. Bruhns and P. Schiesse, βA continuum model ofelastic-plastic materials with anisotropic damage by orientedmicrovoids,β European Journal of MechanicsβA/Solids, vol. 15,pp. 367β396, 1996.
[7] I. Tsukrov andM.Kachanov, βEffectivemoduli of an anisotropicmaterial with elliptical holes of arbitrary orientational distribu-tion,β International Journal of Solids and Structures, vol. 37, no.41, pp. 5919β5941, 2000.
[8] I. Tsukrov and J. Novak, βEffective elastic properties of solidswith defects of irregular shapes,β International Journal of Solidsand Structures, vol. 39, no. 6, pp. 1539β1555, 2002.
[9] R. J. M. Smit, W. A. M. Brekelmans, and H. E. H. Meijer, βPre-diction of the mechanical behavior of nonlinear heterogeneoussystems by multi-level finite element modeling,β ComputerMethods in Applied Mechanics and Engineering, vol. 155, no. 1-2, pp. 181β192, 1998.
[10] P. Ponte Castaneda andM. V. Nebozhyn, βVariational estimatesof the self-consistent type for the effective behaviour of somemodel nonlinear polycrystals,β Proceedings of the Royal SocietyA:Mathematical, Physical and Engineering Sciences, vol. 453, no.1967, pp. 2715β2724, 1997.
[11] P. Ponte Castaneda and J. R. Willis, βVariational second-orderestimates for nonlinear composites,β Proceedings of the RoyalSociety A: Mathematical, Physical and Engineering Sciences, vol.455, no. 1985, pp. 1799β1811, 1999.
[12] D. R. S. Talbot and J. R. Willis, βBounds for the effectiveconstitutive relation of a nonlinear composite,β Proceedings ofthe Royal Society A: Mathematical, Physical and EngineeringSciences, vol. 460, no. 2049, pp. 2705β2723, 2004.
[13] J. Aboudi, βMicromechanics-based thermoviscoelastic consti-tutive equations for rubber-like matrix composites at finitestrains,β International Journal of Solids and Structures, vol. 41,no. 20, pp. 5611β5629, 2004.
[14] J. Hohe and W. Becker, βA probabilistic approach to thenumerical homogenization of irregular solid foams in the finitestrain regime,β International Journal of Solids and Structures, vol.42, no. 11-12, pp. 3549β3569, 2005.
[15] M. Kachanov and I. Sevostianov, βOn quantitative characteriza-tion of microstructures and effective properties,β InternationalJournal of Solids and Structures, vol. 42, no. 2, pp. 309β336, 2005.
[16] J. Fish and R. Fan, βMathematical homogenization of non-periodic heterogeneous media subjected to large deformationtransient loading,β International Journal for Numerical Methodsin Engineering, vol. 76, no. 7, pp. 1044β1064, 2008.
[17] J. Fish,Multiscale Modeling and Simulation of Composite Mate-rials and Structures, vol. 55 of Lecture Notes in Applied andComputational Mechanics, Springer, 2011.
[18] S. Mercier, A. Molinari, S. Berbenni, and M. Berveiller, βCom-parison of different homogenization approaches for elasticvis-coplastic materials,β Modelling and Simulation in MaterialsScience and Engineering, vol. 20, no. 2, Article ID 024004, 2012.
[19] V. A. Levin, K. M. Zingerman, A. V. Vershinin, and M. Y.Yakovlev, βNumerical analysis of effective mechanical proper-ties of rubber-cord composites under finite strains,β CompositeStructures, vol. 131, pp. 25β36, 2015.
[20] A. V. Vershinin, V. A. Levin, K. M. Zingerman, A. M. Sboy-chakov, and M. Y. Yakovlev, βSoftware for estimation of secondorder effective material properties of porous samples withgeometrical and physical nonlinearity accounted for,β Advancesin Engineering Software, vol. 86, pp. 80β84, 2015.
[21] V. A. Levin, V. V. Lokhin, and K. M. Zingerman, βEffectiveelastic properties of porous materials with randomly dispersedpores: finite deformation,β Journal of Applied Mechanics, Trans-actions ASME, vol. 67, no. 4, pp. 667β670, 2000.
[22] V. A. Levin and K. M. Zingermann, βEffective constitutiveequations for porous elastic materials at finite strains andsuperimposed finite strains,β Journal of Applied Mechanics-Transactions of the Asme, vol. 70, no. 6, pp. 809β816, 2003.
[23] V. A. Levin and K. M. Zingerman, βOn the constructionof effective constitutive relations for porous elastic materialssubjected to finite deformations including the case of theirsuperposition,βDoklady Physics, vol. 47, no. 2, pp. 136β140, 2002.
[24] A. I. Lurie, Non-Linear Theory of Elasticity, North-Holland,Amsterdam, Netherlands, 1990 (Russian).
[25] O. C. Zienkiewicz and R. L. Taylor,The Finite Element Method,Volume. 1. The Basis, Butterworth-Heinemann, Oxford, UK,2000.
[26] O. C. Zienkiewicz and R. L. Taylor,The Finite Element Method,vol. 2 of Solid Mechanics, Butterworth-Heinemann, Oxford,UK, 2000.
[27] Fidesys LLC official website, http://www.cae-fidesys.com/.[28] R. Avazmohammadi and P. Ponte Castaneda, βTangent second-
order estimates for the large-strain, macroscopic response ofparticle-reinforced elastomers,β Journal of Elasticity, vol. 112, no.2, pp. 139β183, 2013.
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