XIII. Foundations of Modeling of Composites for Durability

Analysis

Leiting Dong, Beihang University, Beijing, China

Satya N. Atluri, Texas Tech University, Lubbock, Texas, USAIn recent decades, increased use of advanced science and technology, and the wide application of heterogeneous materials, have been experienced in mechanical, aerospace and military industries. For example, metals/alloys with precipitates, and metallic/polymer/ceramic composite materials with fiber/particulate reinforcements are becoming of particular interest. Improvement and accumulation of composite technologies have supported their application as primary structural members of aircrafts, ships, automobiles and other devices, which must sustain high reliability for long durations. Consequently, the establishment of a methodology to evaluate and guarantee the durability of composite materials and development of efficient and accurate tools to model the micromechanical and macromechanical behavior of heterogeneous materials are of fundamental importance. In this chapter, three-dimensional computational grains (CGs, which are “mathematical” or “virtual” grains of polyhedral shape, each containing a heterogeneity) are discussed for micromechanical modeling of heterogeneous materials, following developments in (Dong and Atluri, 2012). Two types of CGs are developed, depending on the heterogeneity in each CG. Each CG can include alternatively a spherical elastic inclusion or a cylindrical elastic fiber. In both cases, an inter-CG compatible displacement field is assumed at each surface of the polyhedral CG, with Wachspress coordinates as nodal shape functions. The Trefftz trial displacement fields in each CG are expressed in terms of the Papkovich-Neuber solutions, which satisfy governing differential equations a-priori. Spherical/cylindrical harmonics are used as the Papkovich-Neuber potentials to derive the Trefftz trial displacement fields. Multi-field boundary variational principles are used for developing the stiffness matrices of the polyhedral CGs. Examples are also given to demonstrate that the CGs can solve three-dimensional micromechanical problems efficiently and accurately. Especially, these CGs can capture the stress concentration around inclusions/fibers, as well as highly accurately predict the homogenized material properties. The rest of this chapter is organized as follows: in section XIII.1, an brief introduction to Micro-Macro modelling with Numerical Methods is presented; in section XIII.2 and XIII.3, a series of computational grains for modeling of particle/fiber composites are presented; in section XIII.4, materials homogenization with CGs is discussed in detail, and parallel computation is implemented to accelerate the analysis; in section XIII.5, we complete this chapter with some concluding remarks.

XIII.1 Introduction to Micro-Macro modelling with Numerical Methods

1.1 Basic Concepts in Micro–Macro Modelling

The macro-scale properties, macro-stress tensor and macro-strain tensor , are defined as the volume average of the micro-scale stress and strain tensors over the RVE

11\* MERGEFORMAT ()where is the volume of the RVE. In the absence of body forces, applying the Gauss theorem allows rewriting these two expressions as

22\* MERGEFORMAT ()The deformation energy at the macroscopic level should be equal to the volume average of micro-scale stress work. It means that at any equilibrium state of the RVE characterized by the stress field σ and the strain field ε, the Hill–Mandel principle must be satisfied, see [1]:

33\* MERGEFORMAT ()From Eqs.1 and 2 the Hill-Mandel principle can be rewritten as:

44\* MERGEFORMAT ()

Without loss of generality, the microscopic displacement field on the RVE boundary can be decomposed into two parts: the mean part and the zero-mean fluctuation

part : 55\* MERGEFORMAT ()

These definitions allow one to express the Hill-Mandel condition in a more convenient form:

66\* MERGEFORMAT ()

The boundary condition of the RVE for the displacement field and the traction field t should be defined in order to satisfy 4 or 6. Three types of boundary conditions are generally considered:1. Constant traction boundary condition: the traction field at the boundary of the

RVE is prescribed in terms of the macroscopic stress: 77\* MERGEFORMAT ()

2. Linear displacement boundary condition: the displacement field at the boundary of the RVE is constrained in terms of the macroscopic strain:

88\* MERGEFORMAT ()

3. Periodic boundary conditions: at the boundary of the RVE, is decomposed into

two parts: a positive part and a negative part with ,

, and with the associated outward normal at

corresponding points and , respectively. The periodic boundary

condition represents the periodicity of fluctuations field and anti-periodicity of the traction field on the RVE boundary:

99\*

MERGEFORMAT ()

For elastic materials, the average tangent moduli can be solved by the linear

relation of the macro-stress tensor and the macro-strain tensor ： 1010\* MERGEFORMAT ()

1.2 Historical Overview and objectives of this chapter

There are several widely-used analytical tools to predict the overall properties of heterogeneous materials. For example, Hashin and Shtrikman (1963) developed variational principles to estimate the upper and lower bounds of the elasticity or compliance tensor. Hill (1965) developed a self-consistent approach to estimate the homogenized material properties. Analytical methods have their unique value in the study of micromechanics. However, because most of these methods follow the work of Eshelby (1957), namely the elastic field of an ellipsoidal inclusion in an infinite medium, these methods can neither consider complex material structures with random distribution of nanoparticles, nor give the nano-scale stress concentrations caused by their interactions. 113Equation Section 3The need for predicting the overall properties of a material with complex geometry, distribution, and arbitrary volume fraction of inclusions, promoted the development of computational tools for micromechanics. A popular way of doing this is to use finite elements to model a Representative Volume Element (RVE). By its concept, a RVE is a microscopic material volume, which is statistically representative of the infinitesimal material neighborhood of the macroscopic material point of interest. By modeling simple loading cases of the RVE, the microscopic stress field

and strain field in the RVE can be computed by the finite element method. And the homogenized material properties are calculated by relating the macroscopic (average) stress tensor to the macroscopic (average) strain tensor. The Finite element method and asymptotic homogenization theory were also combined to perform multi-scale modeling of structures composed of heterogeneous materials. However, it is well known that, primal finite elements, which involve displacement-type of nodal shape functions, are highly inefficient for modeling stress concentration problems. Accurate computation of the fields around a single inclusion or fiber may need many thousands of elements, as shown in Figure 1. Moreover, meshing of a RVE which contains a large number of inclusions/fibers, can be human-labor intensive. For the expensive burden of computation as well as meshing, the above-mentioned computational models mostly use a Unit Cell as the RVE, assuming the microstructure of material is strictly periodic. This obviously cannot account for the complex shapes and distributions of materials of different phases.

Figure 1: The mesh of an Al/SiC Unit Cell model using around 76,000 ten-node

tetrahedral elements with ABAQUS in the study of Chawla, Ganesh [2]In order to reduce the burden of computation and meshing, the authors proposed the idea of Computational Grains. Each Computational Grain can include an inhomogeneity in it and the microstructures are discretized with several Computational Grains. Compared to FEM, the Computational Grains method needs no fine meshes and can directly model the microstructures. Computational Grains can save several orders of magnitude of computational burden and, in addition, precisely capture the stress concentrations (which enable the prediction of damage precursors at interfaces) around microstructural fibers/inclusions much more accurately than the usual FEM. With Computational grains, virtual design and virtual testing of composites containing a large amount of inhomogeneities can be carried out, so composites can be designed, tested and optimized in silico before they are actually manufactured in the laboratory.This chapter is aimed at presenting in detail high-performance computational tools for the modeling of composite materials using Computational Grains, which can directly simulate the micro- and macro-scale mechanical behaviors of composite materials efficiently and accurately. It systematically introduces state-of-the-art computational

methods to develop a high-performance computational tool with a special focus on the Computational Grain method for micromechanical analysis. In the current state of the art, the method of Computational Grains to model the microstructure can be used in conjunction with MultiScale Modeling, to perform a two level analysis of the microstructure first to compute the macrostructural properties, and then to perform the macrostructural analysis using structural finite elements. In reverse, a macrostructural analysis may be performed first using structural finite elements, and then a local microstructural analysis using Computational Grains may be performed, to compute the microstructural interfacial stresses accurately, to predict microstructural damage. With advances in petascale and quantum computing, in the near future it will be possible to model entire structural components using the presently described micromechanical Computational Grains directly rather than using the currently popular structural finite elements. Thus damage precursors at the micro level in a structural component can easily be detected, and the life of a structural component can be more precisely predicted.

XIII.2 Computational Grains for Particulate Composites

2.1 Governing equations for 3D elastic heterogeneous materials

A typical representative material or volume element (RVE) of 3D elastic particulate composites is shown in Figure 2a. Details about algorithms to construct random microstructures and to form the RVE can be found in [3]. Here we just use the algorithms to build the RVE and focus on the CG method itself. Using the meshing technology presented in Appendix A, the RVE is discretized into virtual polyhedral CG elements by tessellation methods that are based on the location and size of heterogeneities, as illustrated in Figure 2b. In each CG, a spherical void/elastic inclusion can be included. Figure 2c is a polyhedral CG with a spherical inhomogeneity.The solutions of 3D linear elasticity for the matrix as well as inclusions in each CG should satisfy the following equations of the equilibrium equation, the compatibility equation, as well as the constitutive relations (assuming that both the matrix and the inclusion are linear elastic and undergo only small deformations):

1212\* MERGEFORMAT ()

1313\* MERGEFORMAT ()

1414\* MERGEFORMAT ()

where the superscript denotes the matrix material, and denotes multiple

inclusions respectively. , , are respectively the displacements, strains and

stresses in the matrix and in the inclusions. is the body force, which can be

neglected for micromechanics of composites. and are the divergence and

gradient operators. and are Lamé constants for

matrix/inclusions, where and are the Young's modulus and Poisson's ratio. is

the 3D unit tensor. The interface conditions can be written as:

1515\*

MERGEFORMAT ()

1616\*

MERGEFORMAT ()The boundary conditions can be written as:

1717\* MERGEFORMAT ()

1818\* MERGEFORMAT

()where and are the prescribed boundary displacements and boundary tractions at

the displacement boundary and the traction boundary of the domain ,

respectively.

(a)

(b)

(c)Figure 2: (a)An illustration of a matrix containing multiple inhomogeneities; (b) the RVE after tessellation into virtual CGs; (c) a representative polyhedral CG with an

inhomogeneity.

2.2 Multi-field boundary variational principles for 3D CG method

In this subsection, multi-field boundary variational principles for the CG method are presented in detail. In the CG method, the constitutive equations, compatibility equations and equilibrium equations are all a-priori satisfied by the assumed Trefftz functions for displacement fields in the matrix and the heterogeneity individually, while the interface/boundary conditions are to be satisfied in a weak sense from the stationarity conditions of a scalar functional. The detailed discussion is given below.In order to develop CGs, we consider independently assumed displacement fields in the interior of each phase, which is a linear combination of complete Trefftz trial functions as discussed in section 3.3. That is to say, the displacement fields satisfy the Navier’s equation in the matrix/ inclusion individually:

1919\*

MERGEFORMAT ()

Another set of displacement , which satisfy the inter-CG displacement continuity

and essential boundary conditions a-priori are introduced:

2020\* MERGEFORMAT ()

Then the inter-CG traction reciprocity as well as the matrix/heterogeneity interface conditions can be satisfied in a weak form by the condition of stationarity of the

boundary functional (note that is a boundary-only functional, since the

equilibrium, compatibility, and constitutive equations are all satisfied by the Trefftz

functions ):

2121\*

MERGEFORMAT ()which leads to Euler-Lagrange equations:

2222\*MERGEFORMAT ()

When the element includes a void instead of an elastic inclusion, is merely

assumed independently at , and we use the following variational principle:

2323\*MERGEFORMAT ()which leads to Euler-Lagrange equations:

2424\* MERGEFORMAT ()

2.3 Papkovich-Neuber solution

For 3D isotropic elasticity, the general solutions in the matrix and in the inclusion can be expressed by using the Papkovich-Neuber solution, see [4]:

2525\* MERGEFORMAT ()

are scalar and vector harmonic functions, which are sometimes called

Papkovich-Neuber potentials.The second equation in 25 can be written in the following index form:

2626\* MERGEFORMAT ()

An interesting fact of the 3D Papkovich-Neuber solution is that, the 3D displacements as in 25 have a very similar form to the displacements in 2D expressed in terms of complex potentials, as shown in [5]. However, unlike the approach of complex potentials, for a specific displacement field in 3D, the harmonic potentials have high degrees of freedom. That is to say, there may exist many different sets of ,

which are harmonic potentials of the same specific displacement field. This renders one to think about if it is possible to drop the scalar harmonic function, to express the solution as:

2727\* MERGEFORMAT ()

It was proved in [4] that 27 is complete for the infinite region which is external to a closed surface for any . However, for a simply-connected domain, 27 is complete only when .

By expressing to be a specific function of , it is shown that a specific case of the

Papkovich-Neuber solution only is:

2828\* MERGEFORMAT ()

which is complete for a simply connected domain, for any .For detailed discussions of the completeness of the Papkovich-Neuber solution, see [4].

2.4 Spherical harmonics

As shown in subsection 3.3.1, in Eq.27 and 28 is a vector which should satisfy the Laplace equation. For spherical inclusion/void problems discussed in this chapter, we express as a linear combination of spherical harmonics. In this subsection, we give a brief introduction to spherical harmonics. For detailed discussions, one can refer to

the monograph [6] and Appendix B. See the spherical coordinates as shown in Figure 3. For the internal problem of a sphere, see Figure 4a, , which satisfy the Laplace equation, can be expanded as:

2929\*

MERGEFORMAT ()For external problems in an finite domain, an hollow sphere for instance, see Figure4b, can be expanded as:

3030\*

MERGEFORMAT ()where

3131\*MERGEFORMAT ()

Figure 4 gives visual representations of the first few real spherical harmonics. Yellow portions represent regions where the function is positive, and blue portions represent where it is negative. The distance of the surface from the origin indicates

the absolute value of in angular direction.

Figure 3: Spherical coordinates.

Figure 4: Illustrations of typical domains for the Laplace equation: (a) a sphere; (b) a finite domain external to a sphere.

Figure 5: Illustrations of spherical harmonics

2.5 The scaled Trefftz trial functions for the inclusion and matrix

Now we can express the Trefftz trial functions used in the CG method in terms of the Papkovich-Neuber solution, in which spherical harmonics are employed. For the

inclusion, as shown in Figure 1c, the displacement field in inclusions can be

derived by substituting the non-singular harmonics:

3232\*

MERGEFORMAT ()into Eq.28:

3333\*

MERGEFORMAT ()

where the superscript denotes the inclusion, and , are the unknown

coefficients in (Eq.39) and (Eq.45).

The displacement field in the matrix is the summation of (the non-singular

part) and (the singular part, with the singularity located at the center of the

inclusion). can be derived by substituting

3434\*

MERGEFORMAT ()

into Eq. 28, and can be derived by substituting

3535\*

MERGEFORMAT ()into Eq. 27:

3636\*MERGEFORMAT ()

where , , , are the unknown coefficients in (Eq.60). and are the

characteristic lengths introduced to scale the Trefftz trial functions, thereby the difficult algebraic problem of solving ill-conditioned systems of equations can be avoided. For this problem, is the maximum radial distance of points in the matrix,

and is the radius of the inclusion. Therefore, and are confined

between 0 and 1 for any point in the matrix, and is confined between 0 and 1

for any point in the inclusion.

Figure 6: Illustrations of the characteristic lengths

We illustrate the reason why we use characteristic lengths to scale the Trefftz trial functions. See Figure 7 for the geometry of the element. Material properties of the matrix are . Two kinds of heterogeneities are considered: an elastic

inclusion with , and a void. Stiffness matrices of the CG are computed,

with and without using characteristic lengths to scale Trefftz trial functions. Condition number of coefficient matrices in Eqs.42 and 46 used to relate , ,α β γ to q

with/without using characteristic lengths to scale Trefftz Trial functions for the element is shown in Table 1. We can clearly see that by scaling the Trefftz functions using characteristic lengths, the resulting have significantly smaller condition

numbers. Elastic Inclusion Void

Characteristic Length

ScaledNot

scaledScaled

Not scaled

Condition number

1.08×103 3.7×1035 2.8×103 2.6×1035

Table 1: Condition number of coefficient matrices used to relate to

Figure 7: An element with a spherical inclusion/void used for the condition number test

2.6 Algorithm for the implementation of CGs

The inter-CG compatible displacement field is assumed at with Wachspress

coordinates as nodal shape functions. Figure 8 shows the Wachspress coordinates for one node of a regular pentagon, which may be the surface of a polyhedral CG.

Figure 8: Wachspress coordinates as nodal shape functions

Using matrix and vector notations, we have:

3737\* MERGEFORMAT ()

The displacement field in and its corresponding traction field at as:

3838\* MERGEFORMAT ()

It should be noted that and are Trefftz trial function matrices, which will be

introduced in the next section.When an elastic inclusion is to be considered, the Trefftz displacement field in the inclusion is independently assumed. We have:

3939\* MERGEFORMAT ()

Therefore, finite element equations can be derived using the following three-field boundary variational principle (Eq.21), which can be written in the matrix form:

pi 4040\*MERGEFORMAT ()

This leads to finite element equations:

4141\*

MERGEFORMAT ()where

4242\* MERGEFORMAT ()This equation can be further simplified by static-condensation:

4343\* MERGEFORMAT()where

4444\* MERGEFORMAT ()

When the element includes a void instead of an elastic inclusion, is merely assumed

at . We have:

4545\* MERGEFORMAT ()

We use the following variational principle50, which can be written in the matrix form:

4646\*MERGEFORMAT ()And corresponding finite element equations are:

4747\*

MERGEFORMAT ()where

4848\* MERGEFORMAT ()

Similarly, this equation can be further simplified by static-condensation.

2.7 Validation of Computational Grains

In order to evaluate the overall performances of mathematical computational grains for modeling problems with inclusions or voids, we consider the following problems: an infinite medium with a spherical inclusion or void in it. Exact solutions can be obtained using Eshelby’s solution and the equivalent inclusion method. For details of the exact solution, see [1].

The material properties of the matrix are . When an elastic inclusion

is considered, the material properties of the inclusion are . The

magnitude of the remote tension in the direction of is equal to 1. The radius of

the inclusion/void is 1. For numerical implementation, the infinite medium is truncated to a finite cube, as illustrated in Figure 9. The length of each side of the truncated cube is equal to . For both the two cases with an elastic inclusion or a void, only one CG is used in the computational model. Traction boundary conditions are applied to the outer-boundary of the element. For each CG, the displacements of

each node on the lower surface are constrained to be the same as the exact solutions.

Figure 9: A spherical elastic inclusion or void under remote tension

First, we compute the eigenvalues of the stiffness matrices of a single CG. This is done in the original and rotated global Cartesian coordinate system. Numerical results are shown in Table 2.As can clearly be seen, the CG is stable and invariant, because additional zero energy modes do not exist, and eigenvalues do not vary with respect to the change of coordinate systems.

Eigenvalues Rotation=0°&45

°inclusion void

1 2.3383 2.26812 1.0878 1.06403 1.0805 1.05644 0.8408 0.86325 0.8408 0.82416 0.8349 0.82417 0.6267 0.62788 0.6161 0.61139 0.6002 0.596310 0.6002 0.596311 0.5530 0.570712 0.5408 0.570713 0.5408 0.560214 0.5029 0.502915 0.2984 0.2994

16 0.2511 0.251817 0.2511 0.251818 0.1915 0.197819 0.1915 0.197820 0.1692 0.172621 0.1219 0.125722 0.0000 0.000023 0.0000 0.000024 0.0000 0.000025 0.0000 0.000026 0.0000 0.000027 0.0000 0.0000

Table 2: Eigenvalues of stiffness matrices of different computational grains when an elastic inclusion/void is considered

We also compare the computed along axis , along axis , to that of the

exact solution. As shown in Figure 10 and Figure 11, no matter an elastic inclusion or a void is considered, computational grains always give very accurate computed stresses, even though only one CG is used.

0 2 4 6 8 10-0.05

0

0.05

0.1

0.15

0.2Analytical solutionCGM

0 2 4 6 8 100.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4Analytical solutionCGM

Figure 10: Computed along axis , along axis for the problem with an

elastic inclusion

1 2 3 4 5 6 7 8 9 10-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7Analytical solutionCGM

1 2 3 4 5 6 7 8 9 10-0.2

0

0.2

0.4

0.6

0.8

1Analytical solutionCGM

Figure 11: Computed along axis , along axis for the problem with a void

XIII.3 Computational Grains for Fiber Composites

A typical representative material or volume element (RVE) of 3D elastic fiber composites is shown in Figure 12a. Details about algorithms to construct random microstructures and to form the RVE can be found in [3]. Here we just use the algorithms to build the RVE and focus on the CG method itself. Using the meshing technology presented in [7], the RVE is discretized into virtual polyhedral CG elements by tessellation methods that are based on the location and size of heterogeneities, as illustrated in Figure 12b. In each CG, an elastic fiber can be included. Figure 12c is a polyhedral CG with a fiber.

(a)

(b)

(c)Figure 12: (a)An illustration of a matrix containing multiple inhomogeneities; (b) the RVE after tessellation into virtual CGs; (c) a representative polyhedral CG with an

inhomogeneity.

3.1 Multi-field boundary variational principles for Fiber Composites

In this subsection, multi-field boundary variational principles for the CG method are presented in detail. In the CG method, the constitutive equations, compatibility equations and equilibrium equations are all a-priori satisfied by the assumed Trefftz functions for displacement fields in the matrix and the fiber individually, while the interface/boundary conditions are to be satisfied in a weak sense from the stationarity conditions of a scalar functional. The detailed discussion is given below. Another set of displacement , which satisfy the inter-CG displacement continuity and essential boundary conditions a-priori are introduced:

4949\* MERGEFORMAT ()

Then the inter-CG traction reciprocity as well as the matrix/heterogeneity interface conditions can be satisfied in a weak form by the condition of stationarity of the

boundary functional (note that is a boundary-only functional, since the

equilibrium, compatibility, and constitutive equations are all satisfied by the Trefftz

functions ):

5050\*

MERGEFORMAT ()which leads to Euler-Lagrange equations:

5151\* MERGEFORMAT ()

where is the interface of the matrix and the fiber.

3.2 Papkovich-Neuber solutions with cylindrical harmonics

In this sub-section, we give a brief introduction to the Trefftz trial functions used in the CG method. For composites containing cylindrical fibers, the Trefftz trial functions are expressed in terms of the Papkovich-Neuber solution, in which cylindrical harmonics are employed.

For the fibers, as shown in Figure 12c, the displacement field can be derived by

substituting the non-singular harmonics:

5252\*

MERGEFORMAT ()into:

5353\*

MERGEFORMAT ()

The displacement field in the matrix is the summation of (the non-singular

part) and (the singular part, with the singularity located at the center of the

inclusion). can be derived by substituting:

5454\*

MERGEFORMAT ()into:

5555\*

MERGEFORMAT ()

and can be derived by substituting

5656\*

MERGEFORMAT ()into:

5757\* MERGEFORMAT

()

Then the displacement field in the matrix can be expressed as:

5858\* MERGEFORMAT ()

where , , , are Bessel functions.

3.3 Stiffness matrix and algorithmic implementation of CGs

For the CGs containing fibers, the inter-CG compatible displacement field is assumed at with Wachspress coordinates as nodal shape functions. Using matrix

and vector notations, we have:

5959\* MERGEFORMAT ()

The displacement field in and its corresponding traction field at as:

6060\* MERGEFORMAT ()

It should be noted that and are Trefftz trial function matrices.

The Trefftz displacement field in the fiber is independently assumed. We have:

6161\* MERGEFORMAT ()

Therefore, finite element equations can be derived using the following three-field boundary variational principle (Eq.50), which can be written in the matrix form:

6262\*

MERGEFORMAT ()This leads to finite element equations:

6363\*MERGEFORMAT ()where

6464\* MERGEFORMAT ()

3.4 Validation of Computational Grains

In order to evaluate the overall performances of mathematical computational grains for modeling problems with inclusions or fibers, we consider the following problems: an infinite medium with a spherical inclusion or a cylindrical fiber in it. Exact solutions can be obtained using Eshelby’s solution and the equivalent inclusion method. For details of the exact solution, see [1].

The material properties of the matrix and the fiber are , ,

respectively. The magnitude of the remote tension in the direction of is equal to

1. The radius of the fiber is 1. For numerical implementation, the infinite medium is truncated to a finite cube, as illustrated in Figure 13.The length of each side of the truncated cube is equal to . For this example, only one CG is used in the computational model. Traction boundary conditions are applied to the outer-boundary of the element. For each CG, the displacements of each node on the lower surface are

constrained to be the same as the exact solutions. We compare the computed along

axis , along axis , to that of the exact solution. As shown in Figure 14,

computational grains always give very accurate computed stresses, even though only one CG is used.

Figure 13: A cylindrical fiber under remote tension

0 2 4 6 8 10-0.05

0

0.05

0.1

0.15

0.2

0.25Analytical solutionCGM

0 2 4 6 8 100.5

0.6

0.7

0.8

0.9

1

1.1

1.2Analytical solutionCGM

Figure 14: Computed along axis , along axis for the problem with a fiber

XIII.4 Modeling of Composites with Computational Grains

4.1 Materials homogenization with CGs

The mesh of an RVE using the CG method is generally nonperiodic so that periodic boundary conditions cannot be directly enforced in a node-to-node fashion. Thus, here we illustrate how the periodic boundary conditions are enforced for CGs without periodic meshing. Figure 15 is a schematic diagram for an RVE after tessellation. For each node at , there is a corresponding point at by projection. One can

locate in a specific polygonal surface of one CG at . And thus, by using the

nodal interpolation function of the Wachspress coordinates, the first equation in Eq.9 can be written as:

6565\*

MERGEFORMAT ()

where is the interpolation function. By applying Eq.65 for every node on of

the RVE, the following constraint equation is obtained:

6666\* MERGEFORMAT ()

where denotes the displacement vector of all nodes.

Figure 15: Illustrations of imposing periodic boundary conditions

After solving the global stiffness matrix equation of a RVE together with the constraints for constant traction, linear displacement, or periodic boundary conditions,

and can be calculated. For elastic materials, the average tangent moduli can

be solved by the linear relation of the macro-stress tensor and the macro-strain

tensor ： 6767\* MERGEFORMAT ()

Up to now, we have discussed in detail the entire set of steps to model particulate composites and porous materials, including the RVE construction (Chapter 12), Voronoi meshing(Appendix A), CG simulation and homogenization. By embedding all these four steps in an in-house CG code, an entirely automatic and highly efficient process of micromechanics simulation and homogenization is realized. One should however notice that, while taking advantages of analytical methods such as the Papkovich-Neuber solution and harmonic functions, the current methodology is very much different from the analytical micromechanics approaches such as the self-consistent method and variational bounds, see [1, 8]. The current methodology consisted of direct numerical simulations by CGs, is in deed a highly-accurate and efficient numerical approach, where random distributions of particles and pores can be easily considered, and stress concentrations in and around heterogeneities can be directly captured, as well as obtaining the homogenized material properties simply through computation.Now we give examples of using CGs for materials homogenization. In the first example, a Unit Cell model of Al/SiC material is considered. The material properties

are: , , , . The volume fraction of SiC is

20%. This model was studied in [2], using around 76,000 ten-node tetrahedral elements with ABAQUS. However, in this chapter, we use just one computational grain, see in Figure 16. As shown in Table 3, although only one CG is used, the homogenized Young’s modulus is quite close to what is obtained by using round 76,000 ten-node tetrahedral elements with ABAQUS.

(a) (b)

Figure 16: The mesh of an Al/SiC Unit Cell model using: (a) around 76,000 ten-node tetrahedral elements with ABAQUS in the study of Chawla, Ganesh [2]; (b) one

computational grain

MethodYoung’s Modulus

(GPa)CG 103.8

ABAQUS 100.0Table 3: Homogenized Young’s modulus using different methods

In the second example, the response surface of (the effective Young’s Modulus)

with respect to and , with different volume fractions of SiC is investigated. In

this example, random microstructures with multiple inclusions are generated and used as RVEs, and the periodic boundary conditions are applied. Three typical RVEs automatically generated, with different volume fractions, as shown in Figure 17.By

performing CG simulations with different values of and , we can obtain the

relation between the effective Young’s Modulus and the modulus of each

constituent material. As can be seen from Figure 18, at specific volume fractions of

SiC, demonstrates significant variations with respect to and .

Figure 17: Three typical RVEs generated by the RVE construction algorithm (Chapter 12) with different volume fractions

Figure 18: Response surface of (the effective Young’s Modulus) with respect to

and , with different volume fractions of SiC

4.2 Parallel computation and direct numerical simulation of

composites

In the previous works for CGs [7, 9, 10], the stiffness matrix of each CG is computed one by one using sequential computation. Although CGs have shown high efficiency in modelling a RVE containing a small number of inclusions[4, 11-13], parallel computation may further reduce computational time when simulating a RVE containing a large number of inclusions. In this study, in order to accelerate the analysis, parallel computation is implemented with the help of Matlab Parallel Computing Toolbox. The flow chart of the parallel algorithm is illustrated in Figure19. The CG analysis starts with input and initialization of the data structure with given CGs, material properties, constraints and loads. The next step is to determine the number of parallel threads. Suppose there are n CGs after tessellation and k parallel threads for computation. Each of first n-1 parallel threads should compute and

assemble the stiffness matrices of CGs ( rounds x to the nearest integer). The

last parallel thread should compute and assemble the stiffness matrices of the rest of the CGs. Finally, we can solve the system of equations with the assembled global stiffness matrix to obtain the nodal displacements.

Figure 19: The flow chart of the parallel algorithm for nano-composite computational grains

In this subsection, examples of directly computing the micro-scale stress distributions and concentrations using CGs are given. Firstly, we study a RVE of Al/SiC material, with randomly distributed spherical SiC particles. The RVE is shown in Figure 20, discretized with currently discussed computational grains. The material properties of Al and SiC are the same as those in the last example. The size of the RVE is 100

μm × 100 μm × 100 μm. A uniform tensile stress of 100 is applied in the

direction. CGs are used to study the microscopic stress distribution in the RVE. The time for modelling 100, 1000 and 10000 spherical nano-inclusions with/without parallel computation are listed in Table 4. Figure 21 shows the computed stress

distributions of for these three cases. While the inclusions are presenting a relative

uniform stress state, the stress distributions in the matrix show high concentration. To be more specific, high stress concentration is observed near the inclusions, in the direction which is parallel to the direction of loading. On the other hand, at the locations near the inclusions, in the direction which is perpendicular to the direction of loading, very low stress values are observed. This gives us the idea at where damages are more likely to initiate and develop, for materials reinforced by stiffer particles. As seen from Table 4, the CPU time needed for the simulation has been significantly reduced after employing parallel computation. Using the CG method with parallel computation, an RVE with 10000 inclusions can be simulated within 50 minutes. Considering the procedure is also entirely automatic, where in the RVE construction, Voronoi tessellation, and CG computation are efficiently executed, a direct numerical simulation of the nano-composite material is indeed achieved. This is probably impossible when the traditional FEM is used, where tens of thousands of simple finite elements are needed to model a single CG [2].

Figure 20: A RVE with spherical inclusions/voids

n 100 1000 10000

without parallel algorithm 360.28s 2886.23s 30819.18s

with parallel algorithm 38.17s 254.32s 2857.74s

Table 4: Time needed for the CG method

(a)

(b)

(c)

Figure 21: Distribution of by the CGs for the problem of a finite matrix

containing (a)100, (b)1000 and (c) 10000 randomly distributed spherical inclusions

XIII.5 Summary

This chapter discussed the development of 3D polyhedral computational grains for a simple and direct numerical simulation for homogenization and micromechanical stress/strain analysis of particle/fiber composites. In this model, the Trefftz displacement trial functions, which satisfy the Navier’s equations, are constructed from Papkovich-Neuber general solutions. Derivation of these functions in terms of spherical/cylindrical harmonics is a major contribution. Algorithms regarding the development of stiffness matrices of CGs, and the enforcement of periodic boundary conditions, are also discussed in detail. Various numerical examples are given to demonstrate the power of the developed CG model. Compared to analytical methods such as the self-consistent method and variational bounds[1, 8]. The current methodology featuring of direct numerical simulations by CGs, can automatically and efficiently model random distributions of particles and fibers, as well as directly and accurately simulate stress concentrations around heterogeneities. These results of interfacial stresses are of paramount importance in studying the initiation of damage near inclusions/fibers.

Reference

[1] Nemat-Nasser S, Hori M. Micromechanics: overall properties of heterogeneous materials: Elsevier, 2013.[2] Chawla N, Ganesh VV, Wunsch B. Three-dimensional (3D) microstructure visualization and finite element modeling of the mechanical behavior of SiC particle reinforced aluminum composites. Scripta Materialia. 2004;51: 161-5.[3] Vaughan TJ, McCarthy CT. A combined experimental–numerical approach for generating statistically equivalent fibre distributions for high strength laminated composite materials. Composites Science and Technology. 2010;70: 291-7.[4] Lurie A. Three-dimensional problems in the theory of elasticity. Theory of Elasticity: Springer; 2005. p. 243-407.[5] Muskhelishvili NI. Some basic problems of the mathematical theory of elasticity: Springer Science & Business Media, 2013.[6] Hobson EW. The theory of spherical and ellipsoidal harmonics: CUP Archive, 1931.[7] Rycroft C. Voro++: A three-dimensional Voronoi cell library in C++. Lawrence Berkeley National Lab.(LBNL), Berkeley, CA (United States); 2009.[8] Willis JR. Bounds and self-consistent estimates for the overall properties of anisotropic composites. Journal of the Mechanics and Physics of Solids. 1977;25: 185-202.[9] Wang G, Dong L, Wang J, Atluri S. Three-dimensional Trefftz computational grains for the micromechanical modeling of heterogeneous media with coated spherical inclusions. Journal of Mechanics of Materials and Structures. 2018;13: 505-29.[10] Eremeyev VA, Lebedev LP. Mathematical study of boundary-value problems within the framework of Steigmann–Ogden model of surface elasticity. Continuum

Mechanics and Thermodynamics. 2016;28: 407-22.[11] Dong L, Atluri SN. Development of 3D T-Trefftz Voronoi Cell Finite Elements with/without Spherical Voids &/or Elastic/Rigid Inclusions for Micromechanical Modeling of Heterogeneous Materials. Cmc -Tech Science Press-. 2012;29: 169-211.[12] Dong L, Atluri SN. T-Trefftz Voronoi Cell Finite Elements with Elastic/Rigid Inclusions or Voids for Micromechanical Analysis of Composite and Porous Materials. Computer Modeling in Engineering & Sciences. 2012;83: 183-219.[13] Dong L, Atluri SN. Development of 3D Trefftz Voronoi Cells with Ellipsoidal Voids &/or Elastic/Rigid Inclusions for Micromechanical Modeling of Heterogeneous Materials. Computers Materials and Continua. 2012;30: 39.