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]
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[21–23] (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
𝑅, 𝑅: radius vector of a particle in initial and currentstates;
𝑢 = 𝑅 −0
𝑅: displacement vector;𝜉𝑖: material coordinates of a particle;𝑥𝑖: spatial coordinates of a particle;𝑒𝑖: basis vectors of reference system;0
q𝑖= 𝜕
0R/𝜕𝜉𝑖, q
𝑖= 𝜕𝑅/𝜕𝜉
𝑖: basis vectors in initial andcurrent states;0
∇ =0
q𝑖𝜕/𝜕𝜉𝑖, ∇ = q
𝑖𝜕/𝜕𝜉𝑖: gradient operators in initial
and current states;𝐼: identity tensor;
Ψ =0∇𝑅 = 𝐼 +
0∇𝑢 = (∇𝑟)
−1= (𝐼 − ∇𝑢)
−1: deformationgradient;0
𝐸 = (1/2)(Ψ ⋅ Ψ∗ − 𝐼) = (1/2)(0∇𝑢 + 𝑢
0∇+
0∇𝑢 ⋅ 𝑢
0∇): the
Green strain tensor;𝐸 = (1/2)(𝐼 −Ψ
−1⋅ Ψ∗−1) = (1/2)(∇𝑢+𝑢∇−∇𝑢 ⋅ 𝑢∇):
the Almansi strain tensor;Δ = detΨ − 1: relative change of volume;∗: sign of transposition;𝜎: true stress tensor;R = (1 + Δ)Ψ
∗−1⋅ 𝜎: the first Piola-Kirchhoff stress
tensor;0
∑ = (1 + Δ)Ψ∗−1 ⋅ 𝜎 ⋅ Ψ−1: the second Piola-Kirchhoffstress tensor;𝑆0, 𝑆: area of RSE in initial and current states;
Modelling and Simulation in Engineering 3
Γ0, Γ: boundary of the RSE in initial and current states;0
𝑁, 𝑁: 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 𝑆0in the initial state (before deformation), we
will solve a certain number of sequences of boundary valueproblems of nonlinear theory of elasticity [24]:
∇ ⋅ 𝜎 = 0
or0
∇ ⋅R = 0
(1)
with boundary conditions
𝑢|Γ0= 𝑟 ⋅ (Ψ
𝑒− 𝐼) . (2)
Each sequence of problems corresponds to a certain typeof boundary conditions (2) and a certain type of effectivestrain tensor𝐸𝑒 over the RSE (i.e., to a certain type of effectivedeformation gradient Ψ𝑒). Different problems within thesame sequence differ by strain values.
Solving each problem of each sequence we will find thefield of the true stress tensor 𝜎. With the knowledge of it wewill calculate the effective stress tensor 𝜎𝑒 by averaging of 𝜎over the area:
𝜎𝑒=1
𝑆∫𝑆
𝜎𝑑𝑆 =1
𝑆∫Γ
𝑁 ⋅ 𝜎𝑅𝑑Γ. (3)
The second equality in (3) was obtained with the helpof divergence theorem and the fact that the RSE is inequilibrium:
Knowing the deformation gradient that was specified in(2), it is possible to calculate the effective Green strain tensorusing the following formula:
0
𝐸𝑒=1
2(Ψ𝑒⋅ Ψ𝑒∗− 𝐼) . (5)
In a linear case, the effective constitutive relations areestimated as a linear dependence of true stress tensor 𝜎𝑒 uponstrain tensor:
𝜎𝑒
𝑖𝑗= 𝐶𝑖𝑗𝑘𝑙𝐸𝑒
𝑘𝑙. (6)
In the nonlinear case, the obtained effective true stresstensor 𝜎𝑒 is used first for calculation of the effective secondPiola-Kirchhoff stress tensor
0
Σ𝑒:
0
Σ𝑒= (detΨ𝑒) (Ψ𝑒)∗−1 ⋅ 𝜎𝑒 ⋅ (Ψ𝑒)−1 . (7)
In this case, the effective properties are estimated asquadratic dependence of the second Piola-Kirchhoff stress
tensor0
Σ𝑒 upon the Green strain tensor 𝐸𝑒:
0
Σ𝑒
𝑖𝑗=(0)
𝐶 𝑖𝑗𝑘𝑙
0
𝐸𝑒
𝑘𝑙+
(1)
𝐶𝑖𝑗𝑘𝑙𝑚𝑛
0
𝐸𝑒
𝑘𝑙
0
𝐸𝑒
𝑚𝑛. (8)
Thus, estimation of the effective properties of the compos-ite in linear case reduces to the calculation of coefficients𝐶
𝑖𝑗𝑘𝑙
(6), and estimation of effective properties in nonlinear case
reduces to calculation of coefficients(0)
𝐶 𝑖𝑗𝑘𝑙 and(1)
𝐶𝑖𝑗𝑘𝑙𝑚𝑛
(8).At that, the following conditions of symmetry are in force
for tensor components 𝐶𝑖𝑗𝑘𝑙
in (6):
(1) 𝐶𝑖𝑗𝑘𝑙= 𝐶𝑗𝑖𝑘𝑙
(due to symmetry of stress tensor);(2) 𝐶𝑖𝑗𝑘𝑙= 𝐶𝑖𝑗𝑙𝑘
(due to symmetry of strain tensor);(3) 𝐶𝑖𝑗𝑘𝑙
= 𝐶𝑘𝑙𝑖𝑗
(due to existence of strain energydensity).
The same conditions are in force for(0)
𝐶 𝑖𝑗𝑘𝑙 in (8). For(1)
𝐶𝑖𝑗𝑘𝑙𝑚𝑛
in (8) the following conditions are in force:
(1)(1)
𝐶𝑖𝑗𝑘𝑙𝑚𝑛
=(1)
𝐶𝑖𝑗𝑙𝑘𝑚𝑛
(due to symmetry of strain tensor);
(2)(1)
𝐶𝑖𝑗𝑘𝑙𝑚𝑛
=(1)
𝐶𝑖𝑗𝑘𝑙𝑛𝑚
(due to symmetry of strain tensor);
(3)(1)
𝐶𝑖𝑗𝑘𝑙𝑚𝑛
=(1)
𝐶𝑖𝑗𝑚𝑛𝑘𝑙
(due to symmetry of multiplication);
(4)(1)
𝐶𝑖𝑗𝑘𝑙𝑚𝑛
=(1)
𝐶𝑗𝑖𝑘𝑙𝑚𝑛
(due to symmetry of the Piola-Kirchhoff tensor).
Considering the above conditions of symmetry, there are
21 independent constants 𝐶𝑖𝑗𝑘𝑙
or(0)
𝐶 𝑖𝑗𝑘𝑙 and 126 independent
constants(1)
𝐶𝑖𝑗𝑘𝑙𝑚𝑛
.
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 “cut out” 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: 𝑋max, 𝑋min, 𝑌max, and 𝑌min. Then the coordinates ofthe RSE center are calculated:
𝑋center =1
2(𝑋max + 𝑋min) ,
𝑌center =1
2(𝑌max + 𝑌min) .
(9)
After that, the model is shifted to put its center to theorigin (i.e., the coordinates of all mesh nodes are decreasedby 𝑋center and 𝑌center). Also, overall dimensions of the model2𝐴 × 2𝐵 are determined by formulas:
𝐴 =1
2(𝑋max − 𝑋min) ,
𝐵 =1
2(𝑌max − 𝑌min) .
(10)
In addition, the accuracy 𝜀 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 element𝑆min. The accuracy 𝜀 is calculated by formula
𝜀 =
3√𝑆min100.0
. (11)
The value of 𝜀 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: |𝑥 − 𝐴| < 𝜀 OR |𝑥 + 𝐴| < 𝜀 OR |𝑦 − 𝐵| < 𝜀 OR|𝑦 + 𝐵| < 𝜀. Here, 𝑥 and 𝑦 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) |𝑥 − 𝐴| < 𝜀;(2) |𝑥 + 𝐴| < 𝜀;
(3) |𝑦 − 𝐵| < 𝜀;(4) |𝑦 + 𝐵| < 𝜀.
If the conditionwith number 𝑛 is true for a node, the nodenumber is added to the list with the number 𝑛.
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 𝑥 =𝐴 is conducted. For each such node 𝑖 a node −𝑖 is found: theclosest one to the point on the edge 𝑥 = −𝐴 that is oppositeto the node 𝑖. In other words, for the node 𝑖 with coordinates(𝐴, 𝑦𝑖) (which lies on the edge 𝑥 = 𝐴) the closest one (from
nodes lying on the edge𝑥 = −𝐴) to the pointwith coordinates(−𝐴, 𝑦
𝑖) is searched for. When such a node is found, the pair
(𝑖, −𝑖) is added to the first list of pairs of nodes.Then the loopis conducted over all nodes of the edge 𝑥 = −𝐴: if the node −𝑖of 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 𝑖 on the edge 𝑥 = 𝑎 and add it tothe first list of pairs.
Similarly, a pass through all nodes of the edge 𝑦 = 𝐵
is conducted. For each such node 𝑗 a node −𝑗 is found: theclosest one to the point on the edge 𝑦 = −𝐵 that is oppositeto the node 𝑗. In other words, for the node 𝑗with coordinates(𝑥𝑗, 𝐵) (which lies on the edge 𝑦 = 𝐵) the closest one (from
nodes lying on the edge𝑦 = −𝐵) to the point with coordinates(𝑥𝑗, −𝐵) is searched for. When such a node is found, the pair
(𝑗, −𝑗) is added to the second list of pairs of nodes. Then theloop is conducted over all nodes of the edge 𝑦 = −𝐵: if thenode −𝑗 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 𝑗 on the edge 𝑦 = 𝐵 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
𝐸11 = 𝑞: strain or compression along the axis𝑋;
(2)0
𝐸22 = 𝑞: strain or compression along the axis 𝑌;
(3)0
𝐸12 =0
𝐸21 = 𝑞: shear in𝑋𝑌 plane;
(4)0
𝐸11 =0
𝐸22 = 𝑞: composition of tensions orcompressions along two axes:𝑋 and 𝑌;
(5)0
𝐸11 = 𝑞;0
𝐸12 =0
𝐸21 = 𝑞: composition of tensions orcompressions along the axis𝑋 and shear in𝑋𝑌 plane;
(6)0
𝐸22 = 𝑞;0
𝐸12 =0
𝐸21 = 𝑞: composition of tensions orcompressions along the axis 𝑌 and shear in𝑋𝑌 plane,
where 𝑞 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 𝑞.
On the basis of effective strain tensor, the effectivedeformation gradient Ψ 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:
(
𝜀11𝜀12
𝜀12𝜀22
)
=1
2[(
𝜓11𝜓12
0 𝜓22
)(
𝜓11
0
𝜓12𝜓22
) − (
1 0
0 1)] .
(12)
Formulas for gradient’s components in an explicit formare as follows:
𝜓22= √2𝑒
22+ 1,
𝜓12=2𝑒22
𝜓22
,
𝜓11= √2𝑒
11+ 1 − 𝜓
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 (𝑖, −𝑖) the relationship withdisplacement is specified in the following form:
𝑢𝑖− 𝑢−𝑖= −2𝐴 (𝜓
11− 1) ,
V𝑖− V−𝑖= −2𝐴𝜓
12.
(14)
For each pair of nodes from the second list (𝑗, −𝑗) therelationship with displacement is specified in the followingform:
𝑢𝑗− 𝑢−𝑗= −2𝐵𝜓
21,
V𝑗− V−𝑗= −2𝐵 (𝜓
22− 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:
𝑆 = 4𝐴𝐵 ⋅ detΨ. (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:
𝜎𝑒=1
𝑆∫𝑆
𝜎𝑑𝑆 =1
𝑆∫Γ
𝑁 ⋅ 𝜎𝑅𝑑Γ
=1
𝑆∑
𝑖
∫𝛾𝑖
𝑁 ⋅ 𝜎𝑅𝑑𝛾𝑖.
(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 𝑞 is plotted:
0
Σ𝑒
𝑖𝑗= 𝛼0
𝑖𝑗𝑞 + 𝛼1
𝑖𝑗𝑞2. (18)
The dependence is constructed using the least squaresmethod, which allows calculating coefficients 𝛼0
𝑖𝑗and 𝛼1
𝑖𝑗.
(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, 𝑞. Effective properties are evaluated as (8).
The relationship between the calculated coefficients 𝛼0𝑖𝑗
and𝛼1𝑖𝑗in (18) for each problem sequence anddesired effective
elastic moduli 𝐶0𝑖𝑗𝑘𝑙
(of the first order) and 𝐶1𝑖𝑗𝑘𝑙𝑚𝑛
(of thesecond order) is as follows:
6 Modelling and Simulation in Engineering
(1)0
𝐸𝑒 = ( 𝑞 00 0) ⇒
0
Σ𝑒
𝑖𝑗= 𝐶0𝑖𝑗11𝑞 + 𝐶1𝑖𝑗1111
𝑞2 =(1)
𝛼0
𝑖𝑗𝑞 +(1)
𝛼1
𝑖𝑗𝑞2;
(2)0
𝐸𝑒 = ( 0 00 𝑞 ) ⇒
0
Σ𝑒
𝑖𝑗= 𝐶0𝑖𝑗22𝑞 + 𝐶1𝑖𝑗2222
𝑞2 =(2)
𝛼0
𝑖𝑗𝑞 +(2)
𝛼1
𝑖𝑗𝑞2;
(3)0
𝐸𝑒 = ( 0 𝑞𝑞 0) ⇒
0
Σ𝑒
𝑖𝑗= (𝐶0𝑖𝑗12+ 𝐶0
𝑖𝑗21)𝑞 + 4𝐶1
𝑖𝑗12𝑞2 =(3)
𝛼0
𝑖𝑗𝑞
+(3)
𝛼1
𝑖𝑗𝑞2;
(4)0
𝐸𝑒 = ( 𝑞 00 𝑞) ⇒
0
Σ𝑒
𝑖𝑗= (𝐶0𝑖𝑗11+𝐶0
𝑖𝑗22)𝑞+(𝐶
1
𝑖𝑗1111+𝐶1
𝑖𝑗2222+
𝐶1
𝑖𝑗1122+ 𝐶1
𝑖𝑗2211)𝑞2 =(4)
𝛼0
𝑖𝑗𝑞 +(4)
𝛼1
𝑖𝑗𝑞2;
(5)0
𝐸𝑒 = ( 𝑞 𝑞𝑞 0 ) ⇒
0
Σ𝑒
𝑖𝑗= (𝐶0𝑖𝑗11
+ 2𝐶0
𝑖𝑗12)𝑞 + (𝐶1
𝑖𝑗1111+
4𝐶1
𝑖𝑗1112+ 4𝐶1
𝑖𝑗1212)𝑞2 =(5)
𝛼0
𝑖𝑗𝑞 +(5)
𝛼1
𝑖𝑗𝑞2;
(6)0
𝐸𝑒 = ( 0 𝑞𝑞 𝑞 ) ⇒
0
Σ𝑒
𝑖𝑗= (𝐶0𝑖𝑗22
+ 2𝐶0
𝑖𝑗12)𝑞 + (4𝐶1
𝑖𝑗1212+
4𝐶1
𝑖𝑗1222+ 𝐶1
𝑖𝑗2222)𝑞2 =(6)
𝛼0
𝑖𝑗𝑞 +(6)
𝛼1
𝑖𝑗𝑞2.
Formulas for calculation of coefficients𝐶0𝑖𝑗𝑘𝑙
in an explicitform are as follows:
(1) 𝐶𝑖𝑗11
=(1)
𝛼0
𝑖𝑗;
(2) 𝐶𝑖𝑗22
=(2)
𝛼0
𝑖𝑗;
(3) 𝐶𝑖𝑗12= 𝐶𝑖𝑗21
=(3)
𝛼0
𝑖𝑗/2.
For calculation of coefficients 𝐶1𝑖𝑗𝑘𝑙𝑚𝑛
it is necessary tosolve the system of 6 linear algebraic equations:
(1) 𝐶1𝑖𝑗1111
=(1)
𝛼1
𝑖𝑗;
(2) 𝐶1𝑖𝑗2222
=(2)
𝛼1
𝑖𝑗;
(3) 4𝐶1𝑖𝑗1212
=(3)
𝛼1
𝑖𝑗;
(4) 𝐶1𝑖𝑗1111
+ 2𝐶1
𝑖𝑗1122+ 𝐶1
𝑖𝑗2222=(4)
𝛼1
𝑖𝑗;
(5) 𝐶1𝑖𝑗1111
+ 4𝐶1
𝑖𝑗1212+ 4𝐶1
𝑖𝑗1112=(5)
𝛼1
𝑖𝑗;
(6) 𝐶1𝑖𝑗2222
+ 4𝐶1
𝑖𝑗1212+ 4𝐶1
𝑖𝑗1222=(6)
𝛼1
𝑖𝑗.
This system is solved analytically, and the solution is asfollows:
𝐶1
𝑖𝑗1111=(1)
𝛼1
𝑖𝑗; 𝐶1𝑖𝑗2222
=(2)
𝛼1
𝑖𝑗; 𝐶1𝑖𝑗1212
= (1/2)(3)
𝛼1
𝑖𝑗;
𝐶1
𝑖𝑗1122= (1/2)(
(4)
𝛼1
𝑖𝑗−
(2)
𝛼1
𝑖𝑗−
(1)
𝛼1
𝑖𝑗);
𝐶1
𝑖𝑗1112= (1/2)(
(5)
𝛼1
𝑖𝑗−
(3)
𝛼1
𝑖𝑗−
(1)
𝛼1
𝑖𝑗);
𝐶1
𝑖𝑗1222= (1/2)(
(6)
𝛼1
𝑖𝑗−
(3)
𝛼1
𝑖𝑗−
(2)
𝛼1
𝑖𝑗).
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]:
⋅ 105MPa [24] were used for steel, and the constants 𝜆 = 0,39⋅ 105MPa, 𝐺 = 0.186 ⋅ 105MPa, 𝐶
3= −0.013 ⋅ 105MPa, 𝐶
4=
−0.07 ⋅ 105MPa, and 𝐶5= 0.063 ⋅ 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 × 5mm. Cross sectionof wire is square; size of square side ranged from 0.25mm to3mm.
𝐶1122 for reinforced plexiglasversus concentration of reinforcing wires.
Let us see graphs for linear coefficients(0)
𝐶1111,(0)
𝐶1122,(0)
𝐶1212
and(0)
𝐶2222 (see Figures 1–4).
The graphs show that the coefficient(0)
𝐶1111 definingthe material behavior during the deformation along theaxis 𝑋 almost linearly depends on concentration of wires
(Figure 1): the greater the concentration, the more(0)
𝐶1111. The
dependence of coefficient(0)
𝐶2222 responsible for the behaviorof the material during the tension along the axis 𝑌 versusthe concentration is similar (Figure 4). Similarly, coefficients(0)
𝐶1212 (Figure 3) defining the behavior of the material under
shear strain and coefficient(0)
𝐶1122 (Figure 2) are dependentupon concentration of wires.
Let us see graphs for nonlinear coefficients(1)
𝐶111111,(1)
𝐶111122, and(1)
𝐶222222 as well (see Figures 5–7).
As can be seen from the graphs, the coefficient(1)
𝐶111111
(which determines the nonlinearity of the material duringtension along the axis 𝑋) increases in modulus with increas-ing of concentration of wires (Figure 5). The coefficient(1)
𝐶111111 (responsible for the nonlinearity during tension alongthe axis 𝑌) 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
𝐶1111 for reinforced plexiglasversus form of reinforcing wires cross section.
nonlinear coefficients (e.g.,(1)
𝐶111122, 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 × 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)
𝐶1111 and(0)
𝐶2222
(see Figures 8 and 9).
The graphs show that the coefficient(0)
𝐶1111 grows mono-tonically and almost linearly as the cross section of thereinforcing wire is “pulled” along the axis 𝑋 (Figure 8). At
that, the coefficient(0)
𝐶2222 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)
𝐶 𝑖𝑗𝑘𝑙 are slightly dependentupon the shape of wires section as shown by calculations.
𝐶222222 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].
As can be seen from the graphs, the coefficient(1)
𝐶111111
increases in modulus monotonically as you “pull” the crosssection of the wires along the axis 𝑋, (Figure 10), and thedependence is close to linear one. At that, the module of
coefficient(1)
𝐶222222 decreases (Figure 12) also monotonically
but nonlinearly: similar to the coefficient(0)
𝐶2222, the coeffi-
cient(1)
𝐶222222 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)
𝐶121212, 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 Young’s modulus 𝐸 =2000MPa and Poisson’s ratio ] = 0.2.Themechanical proper-ties of a matrix are Young’s modulus 𝐸 = 2MPa and Poisson’sratio ] = 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 Λ is a shear strain and 𝑐 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].
Λ
𝑐 = 0.1 𝑐 = 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 𝑥
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
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