Boundary conditions and mesh…

In the FEM analysis, the fiber was assumed to be an orthotropic elastic material and the matrix is assuming to be nonlinear material in which the stress and plastic strain curve was determined by a nonlinear function with four coefficients as

) e (1 R ε R k

σ = + ₀ ^p+ _∞ − ^−bεp (3.2.1)

where k is the yield stress, and b are parameters which can be determined properly from the stress and strain curve by following the suggestion provided in the ANSYS manual [14].

R 0 R_∞

During the FEM analysis, the mechanical behaviors of fiber composite were simulated by considering the representative volume element. The element type was solid-185. In order to characterize the mechanical properties of composites by employing the RVE, the deformation as well as the boundary condition of the RVE needs to be specified properly. In general, the boundary condition was imposed depending on the loading condition and the geometry of the RVE. In this study, we considered the normal stress and the shear stress into the RVE. It is noted that for the applied stresses component , the full model of the RVE need to be accounted for; however, for the other applied stresses, such as , ( ) and ( ), due to symmetric boundary condition, only a quadrant of the REV was taken into account.

In order to describe the appropriate boundary condition according to each loading σ23

σ11 σ₂₂ σ₃₃ σ₁₂ σ₁₃

with easy, the coordinate system as well the dimension of the RVE with square edge packing as shown in Fig. 3.2 were utilized hereafter. It should be noted that the following boundary conditions implemented in our simulation were referred to the literature [2, 3].

(1) Stress component with σ₂₃.

Because of non-symmetry stress field, the deformation at ,x ) 2 direction, respectively. The associated mesh of RVE for three different fiber arrays were shown in Figs. 3.3-3.5, respectively. Base on the characteristic of periodicity, the boundary condition for this case was given as follows

x1 x₂

In order to avoid the rigid body motion, the bottom corners was placed on the rollers hence an additional displacement constrain was

0

Under this stress component field, we only need to analyze quarter of the RVE because of symmetry

where u, v and w respectively to denote the displacement in , and direction. Figs. 3.6-3.8 illustrates the finite element mesh of SEP, SDP and HP. Base on the characteristic of periodicity, the boundary condition for this case was given as follows. In the following derivation, the dimension and coordinate system of the simulation box was shown in Fig. 3.9.

On =0 and

( )

where a₁ and a₂ indicate any two point with other two identical coordinates.

On x₃=0 and x₃ = face c

In addition, in order to eliminate the rigid body motion, an additional displacement constrain was imposed.

0

u(0,0,0)= (3.2.10)

3.3 Comparison the results of GMC, SCMC and FEM analysis

In GMC, due to the lack of shear-coupling, a direct application of a shear load to a fiber composite will cause the inaccurate result. At this section, the results obtained from GMC, SCMC and FEM will be compared and in order to probe the effect of the shear couple in GMC, the fiber composites subject to the transverse loading . The material properties were given in Table 2.1, where the fiber volume fraction was 60% and those four parameters used in the FEM to simulate the matrix properties were list in Table 3.1. Fig. 3.10 illustrated the matrix stress-strain curves used in the GMC, SCMC and FEM models to ensure that all models have the

σ22

same matrix properties. Figs. 3.11 shows the stress and strain curves of the fiber composites with three different fiber arrays obtained from GMC and FEM analysis under the transverse loading. It can be seen that there is some discrepancy between these two approaches which could be caused by the shear coupling effect in the GMC analysis. However, the results obtained from the SCMC analysis are in a good agreement with the FEM analysis as illustrated in Figs. 3.12. Based on the above comparison, it seems that the SCMC model can provide more accurate stress and strain curves of the fiber composites under transverse loading.

On the other hand, the significant drawback in the SCMC model is the convergence problem, which was also observed by other researchers [15]. To understand the degree of the convergence in the GMC and SCMC models, we adopted the two meshes, one is coarse and the other is fine, in our simulation for the fiber composites with hexagonal packing under pure shear loading. The results obtained for the GMC and SCMC models are demonstrated in Figs. 3.13-3.14, respectively.

Apparently, the GMC model exhibits superior convergence property than the SCMC model. Moreover, in some cases, it is difficult to find the convergence solution in the SCMC analysis. In view of the forgoing, the GMC model still posses its advantage in the convergence issue, although its solution in some cases may not be very accurate. From now on, we will continue to employ the GMC model in the investigation of the mechanical behaviors of the fiber composites, even though some defects exist in the model.

Chapter 4 Effect of fiber array on damping behavior of fiber composites In this chapter, the GMC model was extended to calculate the fundamental damping properties of fiber composites with different fiber arrays and the damping properties were then implemented as input in the calculation of the modal damping capacity of composite structures with vibrations [13]. The damping behaviors of rod type as well as plate type composites structure constructed based on different fiber arrays will be taken into account in this chapter.

4.1 Damping characterization using GMC

The fundamental damping capacities of fiber composites in material principal directions were calculated by applying a simple loading on the RVE. The RVE used in the previous section was employed to evaluate the stress and strain states of the fiber composites when they were subjected to simple loading. For example, for the calculation of damping properties in longitudinal direction, the unidirectional composites was applied a loading and then through the GMC analysis, the stress states in the fiber and matrix can be evaluated. Based on the energy dissipation concept that the specific damping capacity of material in vibration was defined as the ratio of the dissipated energy and the stored energy for per circle of vibration [11]

U

ψ= D (4.1.1)

the specific damping capacity of the composites can be expressed in terms of damping properties and strain energy of the constituents, i.e. fiber and matrix, as [12]

m f

m m f f

U U

U ψ U ψ ψ

+

= + (4.1.2)

where ψ_f= specific damping capacity of the fiber ψ_m= specific damping capacity of the matrix U_f= strain energy stored in the fiber

U_m= strain energy stored in the matrix

Thus, the longitudinal damping properties can be calculated from Eq. (4.1.2) directly, once the strain energy as well as the ingredient damping properties was provided. In the fiber composites, the damping behaviors of fiber and matrix were assumed to be isotropic and the corresponding specific damping capacities were listed in Table 4.1.

where the data were measured experimentally [17]. As a result, by introducing a simple loading (simple tension, or simple shear) on the RVE, the strain and stress of each subcell was evaluated respectively from Eqs. (2.1.64), (2.1.65) in which

η

was the overall strain and can be calculated from the constitutive relation of RVE given in Eq. (2.1.67). Moreover, with Eq. (4.1.2), the specific damping capacity of composites in the material directions can be estimated in terms of the damping properties as well as the strain energies of the fiber and matrix phases. Basically the strain energy was computed from the products of the strain and stress states of each subcell associated with either fiber or matrix phases. It is noted that for unidirectional composites, because of the transverse isotropic attribute, only four independent damping properties (ψ₁₁, ψ₂₂, ψ₁₂, ψ₂₃) needs to be calculated.

The damping property of the unidirectional composites with three different fiber arrangements, i.e. square edge packing, square diagonal packing and hexagonal packing, obtained from GMC in conjunction with energy dissipation concept are summarized in Tables 4.2-4.4, respectively. In the calculation, the fiber volume fraction of composites was assumed to be equal to 60%. The damping properties

evaluated based on ANSYS commercial code for the calculation of strain energy were also included for comparison purpose. It can be seen that the specific damping capacity obtained from the GMC analysis are quite closes to those calculated from ANSYS except for properties. The difference in is attributed the fact that GMC model imposed more constrains in the interfacial condition, i.e. the interface traction rate continuity, such that the shear stress as well as the shear strains around the fiber and matrix interface may not be valid.

η23 η₂₃

4.2 Calculation of vibration damping of composite structures

From the GMC analysis together with the energy dissipation concept, we can have the damping properties of unidirectional composites by means of implementing simple loading, such as tension and pure shear on the RVE. However, when the composite structures are adopted for engineering applications, the vibration in general takes place in bending and torsional modes and the damping properties associated with these modes can not be estimated directly from the GMC approach. Here we adopted the two step simulation procedure to predict the damping behaviors of the composite structures with vibration motion. First, the basic material properties of the unidirectional composites, such as , , , et al, were evaluated using the GMC micromechanical model. In the second step, the material properties were considered as global material properties of equivalent element and utilized as inputs in the composites structures for the structural dynamic analysis. In other words, in the structural level, only the material properties of the composites prevail in the analysis. In this study, the structural dynamic analysis was carried out using the FEM approach in which the elements contain the damping and material properties of equivalent element. The detail analytical procedure was illustrated in Fig. 4.1.

E1 ψ₁₁ E₂ ψ₂₂

The modal damping capacity of composite structures can also be derived from

the strain energy dissipation concept. For a linear elastic material, the strain energy stored in a volume element is expressed as

{ } { }

where is the stiffness matrix of composites. The corresponding dissipated strain energy of a volume element can be written in terms of the specific damping capacity in the material principal directions as

[ ]

^C

where indicated the matrix form of damping properties of equivalent elements as shown in Tables 4.2-4.4.

[ ]

^ψ

⎥⎥

Therefore, with the energy dissipation and the strain energy of the equivalent element, the specific damping capacity of a volume element associated with a deformation can be written as follows

{ } [ ][ ] { }

In the finite element analysis, the strain field of a volume element can be expressed in terms of the nodal displacement in conjunction with the shape function as.

{ }

^{ε =}

[ ]

^B

{ }

^d (4.2.6)

a vibration motion, can be regarded as the mode shape of structure representing the relative nodal displacement of the element associated with its natural frequency.

Thus, once the model shapes of the composite structures were determined, the specific damping capacity of the composites structure can be determined by summating the specific damping capacity of each element calculated from Eq. (4.2.7). In the following, the mode shape of the composite structure will be evaluated from structural dynamics analysis together with finite element approach [13].

{ }

^d

From the principle of virtual work, the governing equation for composite structure with dynamic loading were derived by making the virtual work done by externally applied loads equal to the sum of virtual energy caused by inertial, dissipative, and internal forces for any virtual displacement. For a single element of volume V with surface area of S, this relation is written explicitly as

{ } { } { } { }

{ } { } { } { } { } { }

[

^{δu ρ u δu c u δε σ}

]

^dV

dS T δu dV

F δu

V

T T

T

V S

T T

∫

∫ ∫

+ +

=

+

&

&&

(4.2.8)

in which

{

and indicate the body forces and surface tractions, ρ and c denote the mass density and a damping parameter,

}

F

{ }

^T

{ }

^{δu and}

{ }

δε exhibit virtual displacements and their corresponding strains and

{ }

σ is the assumed stress existing in the body prior to virtual strains applied. For a undamped structure with free vibration or with clamp boundary conditions, all of virtual energy caused by the applied loading terms as well as the damping parameter, c were assumed to be zero, and thus the Eq. given in (4.2.8) was deduced as

{ } { }

^{δu ρ udV (e)}

{ } { }

^{δε σdV 0}

V T (e)

V

T +

∫

=

∫

^&& (4.2.9)

Here the superscript (e) designates that the integration is within a volume element. In the finite element method, the displacement field,

{ }

u and strain field

in the element can be represented by the nodal displacement as well as the shape function as

{ }

^ε

{ }

u =

[ ]

N

{ } { }

d u& =

[ ]

N

{ }

d&

{ }

u&& =

[ ]

N

{ }

d&&

{ }

ε =

[ ]

B

{ }

d (4.2.10)

where

[ ]

^N is the shape functions,

{ }

d is nodal displacement which is function of time,

[ ]

^B is the differentiation of shape function

[ ]

^{N , and}

{ }

dt

u& =du ,

{ }

²₂

dt u u&& =d

indicate the velocity and acceleration respectively. Substituting the Eq. (4.2.10) into Eq. (4.2.9) yields

{ }

δd ρ

[ ] [ ]

N NdV

{ }

d

[ ]

B

{ }

σ dV 0

(e)

V

(e)

V T T

T =

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

∫

^&& +

∫

(4.2.11)

By assuming the material is linear elastic,

{ }

σ in Eq. (4.2.11) can be substituted by . In addition, with the assistance of displacement and strain relation, i.e.,

, Eq. (4.2.11) is written as

{ }

^{σ =}

[ ]

^C

{ }

^ε

{ }

^{ε =}

[ ]

^B

{ }

^d

[ ]

m^(e)

{ }

d&& +

[ ]

k ^(e)

{ }

d =0 (4.2.12)

where the element mass matrix

[ ]

^{m (e)} and element stiffness matrix

[ ]

^k(e) is defined

as

[ ]

^=(e)

∫ [ ] [ ]

V T

(e) ρ N NdV

m (4.2.13)

[ ]

k

[ ] [ ][ ]

B C BdV

(e)

V T

(e)⁼

∫

(4.2.14)

Substituting Eq. (4.2.14) into Eq. (4.2.7) yields

{ } [ ] { } { }

d

[ ]

k

{ }

d 2

1

d k 2 d 1 ψ

(e) T

(e) ψ T

(e)= (4.2.15)

where

[ ]

k ^(e)_ψ representing the “energy dissipation stiffness matrix” is written as

[ ]

k

[ ] [ ][ ][ ]

B C ψ BdV

(e)

V T (e)

ψ ⁼

∫

(4.2.16)

For the global response, the structure mass matrix

[ ]

M and the structure stiffness matrix can be derived through the superposition of the element mass matrix and stiffness matrix

[ ]

, respectively by properly assigning each element matrix in the structure matrix depending on the structure node numbering. As a result, the equation of motion for the structure can be written as

[ ]

^K

[ ]

m^(e) k^(e)

[ ]

M

{ }

d&& +

[ ]

K

{ }

d =0 (4.2.17)

Moreover, the natural frequency and mode shape of the composites structures associated with each vibration mode can be evaluated by solving the eigenvalue problem of Eq. (4.2.17). In this study, the eigenvalue and eigenvector of Eq.

(4.2.17) corresponding to the natural frequency and modal shape of the structure, respectively were calculated by Matlab commercial code with “eig” command. It is worthy to mention that in the calculation of the mode shape of the composite structures, the effect of material damping was neglected and only the mass matrix and stiffness matrix were accounted for. From the definition of specific damping capacity, the modal damping capacity of the structure associated to each modal shape can be expressed in terms of the global stiffness matrix

{ }

^Φ

[ ]

K , the global energy dissipation stiffness matrix

[ ]

Kψ and the corresponding modal eigenvector

{ }

^Φ as

{ } [ ] { } { }

_i ^T

[ ] { }

_i

i ψ T i

i Φ K Φ

2 1

Φ K 2 Φ

1

ψ = (4.2.18)

where the index indicates the i

i

^th modal shape. It is noted that the global energy dissipation stiffness matrix is obtained from the superposition of the energy dissipation stiffness matrix given in Eq. (4.2.16). The code for calculating the modal shapes and damping capacity of composite structures is listed in Appendix C.

[ ]

K ψ

4.3 Discussions of the damping capacity of fiber composites with three different fiber arrays

In order to investigate the fiber arrangement effect on the vibration damping of composite structures, the rods and plates constructed with unidirectional composites

were employed for demonstration. Two different boundary conditions, i.e. free-free and free-clamped boundary conditions, were accounted in this study.

4.3.1 Vibration with free-free boundary condition

The modal damping capacities of rod-type and plate-type structures with free-free boundary condition were considered at the beginning. The dimensions of the composite structures used in the simulation were illustrated in Fig. 4.2 where the fiber was assumed in the x-direction. It should be noted that in both structures, the unidirectional fibers could be extended either in the x-direction or in the z-direction to simulate the longitudinal and transverse vibrations. Because of the models were applied with the free boundary, the first six modes were the rigid body motion which was neglected in the model analysis. Figs. 4.3 and 4.4 show the modal shapes of the composite rod with fiber in the x and z directions, respectively. It was shown that for the fiber in the longitudinal direction (x-direction), the first mode is torsion mode which is followed by the bending mode. In contrast, for the rod with the fiber in the transverse direction (z-direction), the first two modes are bending modes and the third one is torsion mode. Tables 4.5 and 4.6 show the first three modal damping capacities of the composite rod structures constructed based on three different fiber arrangements. It can be seen that, no matter what the fiber direction is, the SDP packing always exhibits the highest damping capacity suggesting that the composites with SDP microstructure were easier to dissipated strain energy.

The first three modal shapes for the composite plate with free-free boundary condition are shown in Fig. 4.4. Twisting in the x-direction is the first modal shape and the second one is the bending in the x-direction (fiber direction) and the third mode is the twisting in the z direction (transverse direction). The corresponding damping capacity for the modal shapes is shown in Table 4.7. Apparently, the

composite plate created with SDP also posses the highest damping capacity as compared to the other two cases. As a result, for the composite rod and plate in free vibration, the SDP can provide the superior damping responses than the SEP and HP fiber arrays.

4.3.2 Vibration with clamp-free boundary condition

In addition to the free vibration, the cantilever type vibration, i.e. free-clamped boundary condition, were considered in the study. The clamped end was always in the x-direction and the fiber direction could be either in the x-direction or in the z-direction. Fig. 4.2(a) illustrates the composite rod with fiber in the x-direction and the associated modal shapes are presented in the Fig 4.6. The first one and two modes are bending and torsion modes, respectively and the third one is bending again.

It is interesting to mention that the modal shapes for the unidirectional composites with clamped-free boundary condition are different from those with free-free boundary condition as shown in Figure. The modal shapes for composite rod with fiber in the z direction are shown in Fig 4.7. It was observed that all shapes are in the bending modes. The damping capacities of the composite rods with clamped condition corresponding to two different fiber directions are listed in Tables 4.8 and 4.9. Results show that SDP also demonstrate better damping capacity in the cantilever type vibration.

Again, the plate type structure with one side clamped was examined and the clamped condition was implemented in the x direction as shown in Fig. 4.2(b), where the fiber was assumed in the x direction. The modal shapes for the fiber in the x-direction and z direction were shown in Figs. 4.8 and 4.9, respectively. Moreover, the damping capacities for the plates with fiber in the x-direction and z direction were summarized in Tables 4.10 and 4.11, respectively. Similar to the conclusion in the

rod structure, the plate structure made of unidirectional composites with SDP fiber packing exhibits greater damping properties than the plates established based on the other two fiber arrays.

Chapter 5 Conclusions

The GMC micromechanical model was employed successfully to calculate the thermal residual stress of the fiber composites with different fiber arrays, i.e., square edge packing, square diagonal packing, and hexagonal packing, during the cooling process. Based on the micromechanical analysis, the nonlinear mechanical behaviors of the fiber composites in the presence of the thermal residual stress effect

The GMC micromechanical model was employed successfully to calculate the thermal residual stress of the fiber composites with different fiber arrays, i.e., square edge packing, square diagonal packing, and hexagonal packing, during the cooling process. Based on the micromechanical analysis, the nonlinear mechanical behaviors of the fiber composites in the presence of the thermal residual stress effect

在文檔中 探討不同纖維排列下複合材料之機械行為 (頁 48-0)