Bsme Thesis Jfw

8/17/2019 Bsme Thesis Jfw

1/58

A FINITE ELEMENT ANALYSIS OF DAMAGE ACCUMULATION

IN HETEROGENOUS STRUCTURES

A Thesis

Submitted to the Faculty

of

Purdue University

by

Jonathan Frederick Wenk

In Partial Fulfillment of the

Requirements for the Degree

of

Bachelor of Science in Mechanical Engineering

May 2003


2/58

ii

TABLE OF CONTENTS

LIST OF TABLES............................................................................................................. iii

LIST OF FIGURES ........................................................................................................... iv

ABSTRACT....................................................................................................................... vi

CHARPTER 1: INTRODUCTION .................................................................................... 1

1.1: Motivation............................................................................................................... 1

1.2: Damage in Composite Structures............................................................................ 2

1.2.1: Mechanisms and Accumulation........................................................................ 21.2.2: Damage in Military Applications...................................................................... 4

1.3 Review of Previous Work......................................................................................... 5

1.4: Thesis Statement ...................................................................................................... 7

CHAPTER 2: MODEL CONSTRUCTION ....................................................................... 8

2.1: Model Description ................................................................................................... 82.2: Defining Elements, Real Constants, and Materials ................................................. 9

2.3: Mesh and Boundary Conditions............................................................................. 10

2.4: Simulation of Damage ........................................................................................... 11

2.5: Analysis Types....................................................................................................... 12

CHAPTER 3: PRESENT RESEARCH............................................................................ 13

3.1: Mathematical Representation of Laminated Composites and Loading ................. 13

3.2: Modal Response of Laminated Composite Sheet.................................................. 153.3: Harmonic Forced Response ................................................................................... 19

3.3.1: Results of Damage Case 1 & 2 with Central Loading.................................... 21

3.3.2: Results of Damage Case 1 & 2 with Diagonal Loading................................. 293.3.3: Results of Damage Case 3 with Center and Diagonal Loading...................... 36

3.3.4: Results of Damage Case 4 with Center and Diagonal Loading...................... 38

3.3.5: Results of Damage Case 5 with Center and Diagonal Loading...................... 40

3.4: Force Profile as Applied to Impact Damage .......................................................... 413.5: Discussion of Damage Accumulation, with Examples.......................................... 42

CHAPTER 4: SUMMARY, CONCLUSTIONS, AND RECOMMENDATIONS FOR

FUTURE RESEARCH ..................................................................................................... 48

4.1: Summary and Conclusions .................................................................................... 48

4.2: Recommendations of Future Work........................................................................ 49

LIST OF REFERENCES .................................................................................................. 51


3/58

iii

LIST OF TABLES

Table 2.1: Material Properties of Fiber Reinforced Composite ......................................... 8

Table 2.2: List of Layered Element Types in Ansys........................................................... 9

Table 3.1: Modal Frequency Results ................................................................................ 16

Table 3.2: Description of the Simulated Damage Cases................................................... 20


4/58

iv

LIST OF FIGURES

Figure 1.1: Development of Damage in Composite Laminates (Talreja, 1986)................. 3Figure 1.2: Failure process in laminate composite ............................................................. 3

Figure 1.3: Army Tank with Composite Armor ................................................................ 4

Figure 1.4: Cross-section View of Damaged Armor Composite ........................................ 5

Figure 2.1: Constraint and Load Condition of Baseline Model........................................ 11Figure 3.1: Mode Shape of First Natural Frequency ........................................................ 16

Figure 3.2: Mode Shape of Second Natural Frequency.................................................... 17Figure 3.6: Damage and Force Locations on Composite Sheet, Top View...................... 19Figure 3.7: Layer 1 of Damage Case 1 with Central Loading, Stress vs. Frequency ....... 22

Figure 3.8: Layer 1 of Baseline with Central Loading, Stress vs. Frequency .................. 22

Figure 3.9: Layer 2 of Damage Case 1 with Central Loading, Stress vs. Frequency ....... 23Figure 3.10: Layer 2 of Baseline with Central Loading, Stress vs. Frequency ................ 23

Figure 3.11: Layer 3 of Damage Case 1 with Central Loading, Stress vs. Frequency..... 24

Figure 3.12: Layer 3 of Baseline with Central Loading, Stress vs. Frequency ................ 24Figure 3.13: Layer 14 of Damage Case 1 with Central Loading, Stress vs. Frequency... 25

Figure 3.14: Layer 14 of Baseline with Central Loading, Stress vs. Frequency .............. 25

Figure 3.15: Layer 15 of Damage Case 1 with Central Loading, Stress vs. Frequency... 26

Figure 3.16: Layer 15 of Baseline with Central Loading, Stress vs. Frequency .............. 26Figure 3.17: Layer 16 of Damage Case 1 with Central Loading, Stress vs. Frequency... 27

Figure 3.18: Layer 16 of Baseline with Central Loading, Stress vs. Frequency .............. 27

Figure 3.19: Layer 1 of Damage Case 1 with Diagonal Loading, Stress vs. Frequency.. 30Figure 3.20: Layer 1 of Baseline with Diagonal Loading, Stress vs. Frequency.............. 30

Figure 3.21: Layer 2 of Damage Case 1 with Diagonal Loading, Stress vs. Frequency.. 31

Figure 3.22: Layer 2 of Baseline with Diagonal Loading, Stress vs. Frequency.............. 31Figure 3.23: Layer 3 of Damage Case 1 with Diagonal Loading, Stress vs. Frequency.. 32

Figure 3.24: Layer 3 of Baseline with Diagonal Loading, Stress vs. Frequency.............. 32

Figure 3.25: Layer 14 of Damage Case 1 with Diagonal Loading, Stress vs. Frequency 33

Figure 3.26: Layer 14 of Baseline with Diagonal Loading, Stress vs. Frequency............ 33

Figure 3.27: Layer 15 of Damage Case 1 with Diagonal Loading, Stress vs. Frequency 34Figure 3.28: Layer 15 of Baseline with Diagonal Loading, Stress vs. Frequency............ 34

Figure 3.29: Layer 16 of Damage Case 1 with Diagonal Loading, Stress vs. Frequency 35Figure 3.30: Layer 16 of Baseline with Diagonal Loading, Stress vs. Frequency............ 35

Figure 3.35: Layer 2 of Case 4 with Diagonal Loading, Stress vs. Frequency................. 38

Figure 3.36: Layer 1 of Case 4 with Central Loading, Stress vs. Frequency ................... 39Figure 3.37: Layer 1 of Baseline with Central Loading, Stress vs. Frequency ................ 39


5/58

v

Figure 3.38: Force Profile of an Impact Load, Time and Frequency Based..................... 41

Figure 3.41: Stress Field from the Center Load, Layer Number vs. Frequency............... 44Figure 3.42: Stress Field Generated from the Diagonal Load at the Damage Center....... 45

Figure 3.43: Stress Field from Diagonal Load, Stress vs. Frequency............................... 45

Figure 3.44: Stress Field from the Diagonal Load, Layer Number vs. Frequency........... 46


6/58

vi

ABSTRACT

Wenk, Jonathan Frederick, B.S.M.E, Purdue University, May 2003, A Finite Element

Analysis of Damage Accumulation in Heterogeneous Structures. Major Professor: Dr.Douglas E. Adams, School of Mechanical Engineering

Since structures can be damaged during operation, organizations such as the

Department of Defense care a great deal about how long the material composition of a

structure will last. By basing regular maintenance programs on more reliable information

about the expected life of a structure, safety will be enhanced and costs during the life of

that structure will be reduced. Moreover, the Department of Defense is primarily

interested in the reliability of so-called heterogeneous structures, such as laminated

composites. The focus of this project is to gain a fundamental understanding of the

nature of these laminated composites when subjected to specific damage cases and

ballistic like loads. But the nature of heterogeneous structures is not as intuitive as

homogeneous materials such as aluminum or steel. In order to understand the

progression of the failure modes in a laminated composite, damage models were

constructed and analyzed using Ansys as the Finite Element package. The primary goal

is to be able to predict the progression and accumulation of damage in structures. The

approach was to model a healthy structure and then apply different damage scenarios inreference to the healthy baseline structure in order to evaluate the shift in the stress

distribution. Damage can occur in several layers of a composite, in multiple locations

throughout its volume, and develops through several internal mechanisms. In this project

damage was consistently modeled as the absolute degradation of stiffness within specific

composite layers. The effects of these cases were assessed in terms of the local and

global changes that took place in the structure. Loads, such as those experienced from

ballistic impact, were modeled as distributed loads on the structure’s surface. The stress

field through the thickness of the sheet, as a function of frequency at several locations

throughout the composite, was calculated for several damage scenarios using finite

element modeling. By evaluating the distribution of stress around the damaged regions,

predictions of the progression of damage in the structure were made.


7/58

1

CHARPTER 1: INTRODUCTION

1.1: Motivation

The safety and performance of all commercial, civil, and military structural

dynamic systems deteriorate with usage and time. Current manual inspection and

maintenance scheduling procedures are time consuming, costly, insensitive to small

variations in structural health, and prone to error in severe and mild operating

environments. For these reasons there is an urgent economic and technological need to

develop better methods of evaluating structural integrity and reliability. More

specifically, there is a need to better predict the life of dynamic systems that are

composed of complex materials such as composites. The application of such materials as

load bearing and surface structures has increased significantly over the past 20 years

because composites are both strong and lightweight. Another major attribute of

composite materials is their extraordinary fatigue strength, as compared to homogeneous

materials. In this respect, composites can be exposed to a broad array of load conditions.

A prime example of an extreme case is the use of composites in the armor plating of

military vehicles such as tanks. Materials used in this application must be capable of

withstand substantial transient loads, such as that from a ballistic impact. In order tounderstand the life expectancy of the material, experimental testing and predictive

analysis must be conducted in parallel. The nature of failure in composite materials is far

more complex than in homogeneous materials. This complexity is actually one of the

factors associated with composites resilience under cyclical loading. Unlike metallic

materials where failure modes are usually related with cracking or yielding, laminated

composites have several others, delamination being one of the most frequently

encountered modes (Reddy, Rehfield, Haag, 1984). Thus, a fundamental understanding

of the composition and material properties of composites is necessary. If an accurate

prediction can be made about the reliability of a composite material system after it has

been damaged, then proper maintenance can be administered.


8/58

2

1.2: Damage in Composite Structures

1.2.1: Mechanisms and Accumulation

When structures are subjected to high amplitude or repeated loads, the strength

and integrity of the material can become compromised. This reduction in mechanical

resilience can eventually lead to the failure of the structure over time with usage. In

heterogeneous materials, there are several mechanisms that can lead to the failure of the

structure. Composites can be more sensitive to loading, such as an impact, because they

absorb energy mainly through fracture mechanics rather than elasticity or plasticity

(Cantwell, Curtis, Morton, 1984). As various load conditions are applied to a composite

element, the internal makeup can become damaged with mechanisms such as matrixcracking, delamination, fiber breakage, and local buckling. Primary matrix failure modes

are characterized by cracks that run parallel to the fiber in plies that are not aligned with

the principal tensile loading direction. Secondary matrix failure causes cracks that extend

into adjacent plies, thus initiating delamination. A delamination, also called debonding,

is a crack that extends within the resin rich (matrix material) interface between plies that

may contain different fiber orientations. It has been observed that delamination only

occurs in the presence of matrix cracks. As delamination damage accumulates the

material characteristics change until ultimately the structure fails in the form of fiber

breakage (Talreja, 1986). Figure 1.1 shows the characteristic progression of damage in a

laminated composite. In the locations where failure begins to occur, the stiffness is

reduced and the surrounding material must carry the load. Because the fibers are the load

bearing material within a composite, when they fail the structure is permanently

compromised. Failure takes place in stages, where one damage event can lead to a

sequence of failures inside of the material as shown in the stress-strain curve in Figure

1.2. Failures within a ply are referred to as intralaminar and failures between layers are

referred to as interlaminar. Each plateau in Figure 1.2 is a basic representation of a

failure within a ply in a composite sheet. As consecutive ply failures occur, the end result

is an overall failure of the composite laminate. The figure implies that the failure within


9/58

3

a ply is sudden. But in reality the failure is progressive as mechanisms such as matrix

cracking occur gradually rather than suddenly (Sun, 2003).

Figure 1.1: Development of Damage in Composite Laminates (Talreja, 1986)

Figure 1.2: Failure process in laminate composite


10/58

4

Structural failures in composite materials can be initiated by several means. They

can occur as manufacturing anomalies during initial construction. In some cases the

failure is gradual due to load conditions such as low amplitude fatigue. Failure can also

be immediate and catastrophic from high energy loading such as impacts. In most

gradual cases, the overall strength is not compromised until a significant number of load

bearing fibers have failed.

1.2.2: Damage in Military Applications

The military is utilizing composite materials as armor plating in robust vehicles,

such as the tank shown in Figure 1.3. These vehicles can experience high transient loads

from ballistic impacts, but are also exposed to cyclic loading during use in the field. Instructural composites impact damage may not cause immediate failure. This damage

may, however, give rise to subsequent failure under service loads (Cantwell, Curtis,

Morton, 1984). An example of the severe damage that can occur in ballistic loaded

structures is shown in Figure 1.4. The focus of this project is on the response of

composite sheets as they are exposed to large distributed loads. The result of these loads

is the complete elimination of load bearing capability in that region. For armor on a tank,

this loss in residual strength can lead to a catastrophic event.

Figure 1.3: Army Tank with Composite Armor

Armor and weapon systemsundergo ballistic impacts


11/58

5

Figure 1.4: Cross-section View of Damaged Armor Composite

1.3 Review of Previous Work

Understanding the effects of damage on composites has been the focus of many

studies in the engineering community. Only a few of those studies are mentioned here.

Engblom and Havelka (1991) used a combined analytical and experimental approach to

develop models of damaged composite structures. The experimentation was intended to

quantify the effects of the four major damage mechanisms on the variations in stiffness

and damping characteristics. The data gained from the experiments was used to improve

existing finite element based damage models, as well as improve predictions of changes

in material properties. The result of the research indicates the effects of delamination andmatrix cracking can greatly affect the residual properties and dynamic characteristics of

laminated composites. In order to differentiate between the various failure mechanisms,

failure criteria, developed by other researchers, were incorporated in the assessment of

the damage. Hashin (1980) developed a stress based failure criteria to distinguish

between fiber and matrix failure modes. Lee (1982) used a similar stress based criteria to

distinguish delamination from other failure modes.

Yen, Cassin, Patterson, and Triplett (1997) conducted a progressive failure

analysis of thin walled composite tubes under low energy impact. This study was also a

comparative study of experimental and analytical analyses. The failure criteria developed

therein was integrated into an explicit dynamic analysis code for failure prediction of the

composite tubes. The results provided a good correlation with experimental data of

Delamination and

debonding


12/58

6

impact force histories and certain critical damage modes. Also, the code predicted

nonlinear behavior due to the progression of local damage, within the macroscopic

continuum.

Talreja (1986) developed an internal state variable model, which was used to

determine the overall stiffness properties and intensity of damage in individual modes.

This relationship shows that intralaminar damage reduces all the elastic moduli for

damage with general orientations and changes the initial orthotropic symmetry of a

laminate. But the interlaminar damage does not change the symmetry, only the moduli.

This model represents the effects of matrix cracking on stiffness reduction by evaluating

the development of crack size within the ply. Then, the model characterizes the damage

modes as vectorial quantities that incorporate the geometrical properties of the damage

entities.

Cantwell, Curtis, and Morton (1984) conducted low energy impact and fatigue

testing on carbon fiber laminates. It was found that the effects of impact reduce the

residual strength of the laminate. The effects of various impact energies were studied, as

well as the effects of fatigue on the impacted samples as a further measure of residual

strength. Efforts were made to show that the integration of woven fabric into the stacking

sequence, in place of individual alternating plies, significantly improved the materials

impact performance. The stacking sequence plays a major role in determining the impact

resistance of a composite panel.

Reddy, Rehfield, and Haag (1984) performed experimental and analytical

analyses of laminated composite sheets with prescribed delaminations of know shape and

size in order to assess the effect of such flaws on the structural behavior. Two different

finite element models were constructed to model the delamination. The first simply

reduced the number of beam elements used to connect the plies together. The second,

that is most similar to this project, decreased the modulus of affected elements within the

model. Thus weakening the structure in those areas. The results of the finite element

model provide conservative estimates of behavioral degradation when compared to the

experimental results.


13/58

7

1.4: Thesis Statement

It can be very beneficial to control the location and extent of damage using finite

element analysis when evaluating the performance of a structure under loading.

In this research, the effects of various damage cases, load input locations, and

frequency spectra on the dynamic response of a laminated composite sheet will be

presented. Material properties and dynamic characteristics will be used to generate a

realistic finite element model with which to run the analyses. The results of this model

will be used to provide insight into the progression and accumulation of damage within

the plies of the composite sheet. By evaluating the response of the model using

information about future dynamic loading, predictions will be made with regard to the

internal stress distributions.


14/58

8

CHAPTER 2: MODEL CONSTRUCTION

Before constructing the finite element model of the composite material, time was

dedicated to learning the abilities and functions of Ansys, the software used for the

project. Prior use of other finite element packages proved valuable in learning Ansys, but

modeling composite materials is more difficult than modeling isotropic materials such as

steel. The Help Index and Tutorials were the primary resource in becoming familiar with

the layout of Ansys.

2.1: Model Description

The goal of this project is to analytically model a composite sheet, similar to those

tested by the Department of Defense, and apply simplified ballistic loads to the sheet.

Using dimensions and material specifications from test specimens and reference texts, a

finite element model was constructed in Ansys. The model was based on a 20-ply

composite sheet with S2 Glass/Epoxy for the fiber and matrix material in each of the

plies. The laminated sheet is symmetric about the mid-plane with a stacking sequence of

[(0/90)10]s. This sequence means that the plies alternate between zero and 90 degrees

relative to the fixed global coordinate system. The overall dimensions used for this

model are 4ft x 3ft x 0.25in, which is 1.219m x 0.914m x 6.35mm in metric. The

thickness of each ply is uniform at 0.0125in or 0.3175mm. The material properties (Sun,

2003) used in modeling the sheet are listed in Table 2.1.

Material ExGPa

EyGPa

EzGPa

uxy

GPa

uyz

GPa

uxz

GPa

GxyGPa

GyzGPa

GxzGPa

sten

GPa

r

g/cm3

S2

Glass/Epoxy

43.3 12.7 12.7 0.29 0.5 0.29 4.5 2 4.5 1.7 1.8

Table 2.1: Material Properties of Fiber Reinforced Composite


15/58

9

2.2: Defining Elements, Real Constants, and Materials

There are several steps that must be taken in order to properly define the

composition of a composite structure within Ansys. First is the selection of the element

type. Ansys offers five element types, which are listed in Table 2.2, which can be used to

define layered composites.

Type Name Description Name Description

Shell99 Linear Layered Structural Shell 8-node, 3-D shell, 6-DOF per node,

up to 250 layers

Shell91 Nonlinear Layered Structural Shell 8-node, 3-D shell, 6-DOF per node,

up to 100 layers

Shell181 Finite Strain Shell 4-node, 3-D shell, 6-DOF per node,

up to 255 layers with large strain

Solid46 3-D Layered Structural Solid 8-node, 3-D solid, 3-DOF per node,

up to 250 layers

Solid191 Layered Structural Solid 20-node, 3-D solid, 3-DOF per

node, up to 100 layers

Table 2.2: List of Layered Element Types in Ansys

Due to the simple geometry of the model and the assumption of linear response in

this project, the Shell99 elements were selected. These elements are designed to model

thin to moderately thick plate and shell structures, with a side-to-thickness ratio of 10 or

greater. It is this fact that allowed for the generation of a moderately fine mesh, as

opposed to a very fine mesh that takes longer to solve. Once the element type has been

selected the material properties, layer orientation, and layer thickness must be defined

within each element. In Ansys these properties are set using real constants. Real

constants are user-defined element characteristics, which represent the configuration of

the element. The first step is to define the materials that will be used in the model.

Possible materials are accessed through the material models section of Ansys. In this


16/58

10

dialog window, the orthotropic mechanical properties can be set for any number of

materials. In the real constant dialog window, the preliminary option is to define the

number of layers in the model. Once the number of layers is specified, the composition

of each individual layer is defined. Each layer can be represented by any one of the

material models that have been defined. The layer orientation is defined as the direction

of the layer coordinate system relative the global coordinate system, and the orientation is

defined by entering the angle between the x-axes of each coordinate system. Finally the

thickness of each individual layer can be defined to meet the specifications of the

composite. There may be as many real constants as are necessary to accurately represent

the structure being modeled.

2.3: Mesh and Boundary Conditions

When using Shell99 elements, the mesh density is governed by the side-to-

thickness ratio requirement, which is ten. In this model, the length was split into 26

elements and the width was split into 20 elements. This resolution provides an element

side length of roughly 0.0463m. With the thickness of the sheet at 6.35mm, the actual

side-to-thickness ratio of the model is approximately 7.6. This ratio is below the

specification of 10, but was used anyway to allow for a slightly finer mesh that would be

more sensitive to the response of the model. If there were an application in which the

mesh density needs to be finer, then the use of Solid46 elements would be more suitable.

These elements are typically used for thicker plate structures.

Several types of constraints and loads were applied to the model. In order to

assess the response of the composite model under various conditions, the loads and

constraints were assigned in different configurations. In each of the simulations, all of

the edges of the plate were constrained in all six degrees of freedom. This was done to

emulate the boundary effects if a piece of a composite were taken out of a larger,

continuous part. The force on the plate was applied as a distributed load over an area of

0.0247m2 (45 nodes) with a value of 50N per node. The total force on the plate is 2250N,

which is approximately 505lbf. This value is a reasonable estimate of loading from a

ballistic impact. A comparison was done to evaluate the dynamic effects of a point load,


17/58

11

instead of a distributed load. A point load is able to excite more of the natural modes in a

structure, but the stress effects were unrealistic. The baseline boundary conditions are

shown in Figure 2.1. The load was applied in both the center of the plate and in locations

diagonal to the center. The re-location of the loading distribution was done to excite the

natural modes of the plate and avoid over constraining the plate in all of the simulations.

The loading conditions used in each of the simulations are outlined and discussed in

Chapter 3.

Figure 2.1: Constraint and Load Condition of Baseline Model

2.4: Simulation of Damage

The method used to simulate damage in the composite model was to assume that

the modulus and shear modulus were nearly zero at the location of damage. In order to


18/58

12

avoid singularity issues during calculation, the properties were not set exactly to zero.

For all damage cases, the material properties assigned to that location were as follows:

• Ex= Ey = Ez = 100Pa

• Gxy= Gyz = Gxz = 50Pa

• uxy= uyz = uxz = 0.3

These properties are significantly less than those of a healthy structure. The strength in

damaged regions is decreased by a factor of 108, which essentially acts as zero. These

properties represent the effect of a total failure of the load carrying capability of that

region. The initial effects of damage are difficult to model. For example no stiffness

reduction is assumed after primary matrix failure occurs. This is because transverse

matrix cracks alone usually do not have a significant effect on the laminate stiffness.

Damage characteristics were set in Ansys by using several options. First, the

material properties were created in the material models. This material definition was then

entered into a new real constant at the specific layers in which damage was desired. In

this project, the layer orientation and layer thickness were held constant in the damaged

cases. Then, the damaged region of elements was assigned the characteristics of the new

real constant. This task is accomplished in Ansys by modifying the element attributes in

that region and re-assigning the real constant number. By using this methodology,

damage can be applied in exact locations, at any element on the sheet and in any ply.

This technique is similar to that used in previous research (Yen, Cassin, Patterson, and

Triplett, 1997).

2.5: Analysis Types

Several types of analysis were conducted to assess the structural performance of

the composite sheet. Modals were performed to discover the first five natural frequencies

and mode shapes of the model. A static analysis was done to find the maximum

deflection due to the applied distributed load. The main analysis used to assess the

effects of damage was the harmonic forced response. Several damage scenarios were

simulated, including the re-location of the applied force and damaged regions. The

results of these analyses are outlined in Chapter 3.


19/58

13

CHAPTER 3: PRESENT RESEARCH

3.1: Mathematical Representation of Laminated Composites and Loading

The response and characteristics of laminated composites are governed by many

equations. A laminated plate with a thickness that is small compared to the lateral

dimensions has displacement components u, v, and w that are functions of x, y, and z.

These equations are expanded in terms of the power series of z. Because the

displacements are small, only the first two terms in the series are retained. The first terms

are identified as the displacement components of the mid-plane, the second terms

represent the linear relationship with the deformation in the z direction:

),(),,(

),(),(),,(),(),(),,(

0

0

0

y xw z y xw

y x z y xv z y xv y x z y xu z y xu

yx

x

=

+=

+=

ϕϕ

(3.1)

In general, the stress-strain relation, {σ} and {ε}, of a lamina (ply) is governed by the

elastic compliance matrix [S], which is the inverse of the elastic constant matrix [C]:

{ε}=[C]-1{σ}=[S]{σ} (3.2)

Where the matrix [S] is determined by the mechanical properties of the lamina:

(3.3)


20/58

14

The stress-strain relation for orthotropic materials is simplified due to the

symmetric property of the compliance matrix and follows the relationship:

Although the strains are continuous over the thickness of the laminate, the stressesin the laminae are discontinuous across the interfaces due to the different material

properties resulting from different fiber orientations. For the k th lamina, the stress

components are given by (Sun, 2003)

In equation 3.5 [Q] represents the reduced stiffness, a term similar to the elastic

constants, which is the inverse of the elastic compliance. The strains are described by the

in-plane strain and curvatures due to bending in the sheet. Typically, the strains

associated with bending have the most significant effect. In composites this is influenced

by the stacking sequence of the laminae.

When evaluating the strength within a lamina, there are several criteria that can be

used. One such criterion is the Maximum Stress Criterion, which compares the

maximum tensile strength in the fiber direction, transverse to the fiber direction, in-plane

shear strength to the state of stress found in the structure.

(3.4)

(3.5)


21/58

15

Another important factor to be taken into account are the effects of fatigue on a

structure. The following equation is taken from the Modified-Goodman model of fatigue

life, where the stress in the structure is represented by a mean and alternating value:

f f

a

ut

m

N S S

1

=+

σσ

(3.6)

In terms of characterizing the force input to the system, for this project the force is

assumed to be from a ballistic impact. The desired force input or time of contact can be

found from the impulse-momentum equation:

vmt F ∆=δ (3.7)

Assuming a perfectly elastic collision, the time of contact can be found by rearranging

the equation 3.7:

F

mvt 2

2=δ (3.8)

Where m represents the mass of the projectile, v represents the velocity, and F is the force

it imparts. The use of this equation is discussed further in section 3.4.

3.2: Modal Response of Laminated Composite Sheet

The first step in evaluating the response of the laminated composite sheet was to

perform a modal analysis. To remain consistent, the plate was set with the same

boundary conditions that were used for the harmonic forced response. All four edges of

the sheet were fully constrained in all degrees of freedom. The first five natural

frequencies and modes shapes were found for the healthy sheet. The frequencies are

listed in Table 3.1 and the mode shapes are shown in Figures 3.1 through 3.5.

Reliability and damage accumulations are based, primarily, on low frequency

events. This critical role of low frequency response was the motivation for assessing

only the first five modes of the composite. In some cases, the higher modes do have a

contribution to the magnitude of the response in the lower frequencies. In this project,

those contributions were not taken into consideration.


22/58

16

Mode Number Frequency (Hz)

1 36.97

2 62.83

3 87.22

4 106.16

5 108.13

Table 3.1: Modal Frequency Results

Figure 3.1: Mode Shape of First Natural Frequency


23/58

17

Figure 3.2: Mode Shape of Second Natural Frequency

Figure 3.3: Mode Shape of Third Natural Frequency


24/58

18

Figure 3.4: Mode Shape of Forth Natural Frequency

Figure 3.5: Mode Shape of Fifth Natural Frequency


25/58

19

3.3: Harmonic Forced Response

In order to perform the harmonic forced response it was important to know the

natural frequencies and mode shapes so that the force input and damage could be applied

in the most effective locations. For this project, there were five damage cases that were

evaluated with the load applied at two different locations. The distributed force described

in section 2.3, was applied either at the center of the sheet or diagonal to the center in the

upper left corner of the sheet. The damage and force locations are shown in Figure 3.6.

In this figure, the squares represent the regions in which damage occurs, and the circles

represent the distributed forces.

Figure 3.6: Damage and Force Locations on Composite Sheet, Top View

Damage Case

1, 2, and 5

Damage Case 2Damage Case 3

Damage Case 4

Load 1Load 2


26/58

20

The damage simulated within the composite layers was based on research of laminates in

which the failure modes that occurred were primarily sub-surface. This is typically the

result that forms from impact loading. Each of the damage cases is outlined in Table 3.2.

The results of each of the damage cases are presented in the following section. All

results were found within a frequency range of 30 to 115 Hz. This range was determined

by the first and last natural mode of the sheet.

Damage Case Description of Damage Cases

1 A 4x4 region of elements in the top right corner with layers 4à13

failed, and layers 1à3 and 14à20 healthy

2 Two 4x4 regions of elements in the top right corner and bottom left

corner with layers 4à13 failed, and layers 1à3 and 14à20 healthy

3 A 6x6 region of elements in the top right corner with layers 4à13


4 A 4x4 region of elements in the center of the sheet with layers 4à13


5 A 4x4 region of elements in the top right corner of the sheet with

layers 4à17 failed, and layers 1à3 and 18à20 healthy

(Deeper failure, with more layers)

Table 3.2: Description of the Simulated Damage Cases

When the model was prepared to run in the harmonic forced response, several key

factors were set. The force was applied as a step load to ensure that the load amplitude

was maintained for all of the sub-steps in the frequency range. The load was applied with

a zero phase angle. Also, a constant damping ratio of 0.1 was applied to the entire model.

The geometry of the damage regions was created as squares due to limitations in themesh density of the model. For this reason, the stress was not evaluated in the corners of

the damage regions, due to stress concentration that can occur in sharp geometries.


27/58

21

3.3.1: Results of Damage Case 1 & 2 with Central Loading

The first set of damage cases were conducted with the loading at the center of the

sheet. By applying the load in this location, the natural modes that are associated with

displacement at the center node of the sheet are excited. In this particular application,

modes 1 and 5 are the most susceptible. The following figures show the magnitude of

stress versus frequency from selected layers within the composite. These are compared

to the results of a healthy baseline sheet with the same loading. The plots are generated

from data calculated at the same node in the center of the damaged region. The

presentation of the plots is on an individual basis, due to the limitations in the output

capabilities of Ansys.


28/58

22

Figure 3.7: Layer 1 of Damage Case 1 with Central Loading, Stress vs. Frequency

Figure 3.8: Layer 1 of Baseline with Central Loading, Stress vs. Frequency


29/58

23




30/58

24




31/58

25




32/58

26




33/58

27




34/58

28

Figures 3.7 through 3.12 are the results for the first three layers of the composite,

which are located just above the damage. Figures 3.13 through 3.18 are the results of the

three layers just below the damaged region. The loading that is applied to the sheet is

normal to the surface, which induces bending stress in the sheet. The overall trend of the

stress, as it is distributed through the layers, is that it decreases from the outer surfaces to

the mid-plane. But due to the damage, the stress in the first three layers increases

because of the reduction in the local cross-sectional area. The stress in the three layers

just below the damage decreases with respect to the baseline. The remaining layers

below the damage are not shown, as the magnitude of stress is equivalent to that of the

baseline. Also, the peak stress in bottom four layers do not exceed that of the top three

layers, so the results are not presented here.

Figures 3.7 through 3.18 are derived from the results of the first damage case, but

are also representative of the second damage case. Upon review of the results from the

second case, the response due to this load input is the same at this location through the

thickness of the sheet. The main difference is the distribution of stress in the area that

surrounds the damage, as opposed the area above and below. As the sheet is viewed from

the top, the view shown in Figure 3.6, the stress distribution in the region above and to

the right of the damage is the same for case 1 and 2. At these locations the stress is

higher than the baseline in layers 1 through 10 and is the same in layers 11 through 20.

These regions are between the damage and the edge constraints, thus the increased stress

in the top layers at these locations could be caused by the boundary conditions. The main

difference in the stress distribution for case 1 and 2 is in the regions below and to the left

of the damage. In case 2, the stress is higher than case 1 in all of the layers in these

regions. The magnitude of stress in this region exceeds that found in the center of the

damage region, for both cases. This is caused by the reduction in stiffness in the second

diagonal region. As the load is applied to the sheet, the material is pulled toward the

center. In the second damage case the material around the damage in the top right corner

has weakened support from the left side of the plate. This increases the deflection in the

sheet, thus increasing the strain and stress.


35/58

29

As the loading path changes in the composite, due to ply failure, the stress will

rise and the damage will progress further. In the first case the damage will progress up,

into the layers above the original damage, as well as through the top layers toward the

constraints. In the second case the damage will progress toward the load, as well as the

layers above the original damage.

Besides changes in the magnitude of stress in the sheet, there is also a slight shift

in the frequency at which the peak stresses occur. Modals were also conducted on the

damaged sheets, in order to account for the shift in frequency caused by the damage.

Mode 4 was the most significantly affected, due to the location of excitation in this mode.

For the first damage case, the frequency at this mode shifted down 1 Hz. For the second

damage case, the frequency shifted down 2 Hz. The other modes were mildly affected.

3.3.2: Results of Damage Case 1 & 2 with Diagonal Loading

The second set of damage cases were conducted with the load applied at a

location diagonal to the center of the plate, as indicated in Figure 3.6. By applying the

load in this location, the natural modes that are associated with displacement in the top

left corner of the sheet are excited. In this particular application, modes 2 and 4 are the

most susceptible. The following figures show the magnitude of stress versus frequency

from selected layers within the composite. These are compared to the results of a healthy

baseline sheet with the same loading conditions. The data is taken from the center node

of the damaged region, same as the data from center-loaded cases.


36/58

30

Figure 3.19: Layer 1 of Damage Case 1 with Diagonal Loading, Stress vs. Frequency

Figure 3.20: Layer 1 of Baseline with Diagonal Loading, Stress vs. Frequency


37/58

31




38/58

32




39/58

33




40/58

34




41/58

35




42/58

36

For damage cases 1 and 2, with the load applied in the top left corner, the

frequencies at which the peak stresses occur have shifted. The trend of the stress

distribution through the layers is similar to those noted for the center loading cases. This

load also puts the sheet in bending, which causes the stress to decrease between the outer

surface and the mid-plane of the sheet. The stress in the top three layers is higher in both

cases as compared to the baseline. The stress in the layers below the damage is less than

that found in the baseline, but as the layers progress toward the very bottom the

magnitude of the stresses are equal to that of the baseline. In these cases, when the plate

is viewed from the top, the stress to the left of the damage is decreased in the top five

layers and is the same as the baseline in the remaining layers. This indicates that the

loading path for this input force is through the layers above the damage and between the

damage and the edge constraints. These will be the areas in which the damage progresses

into a worse state.

In comparing the results from the central loaded sheet and the diagonal loaded

sheet, the stresses are within the same order of magnitude. The maximum stress occurs

around 37 Hz in the central loaded cases, with a magnitude of about 72 MPa. But, it

should be noted that around 107Hz, the stress is about 66 MPa. In the diagonal loaded

sheet, the maximum stress occurs around 63 Hz with a magnitude of about 60 MPa, and

at 106 Hz the stress is about 50 MPa. To avoid high fatigue stress, the first and forth

modes of the sheet should not be excited if damage is located in the either of the

excitation locations. Therefore, the central load is more detrimental to the health of the

sheet, in terms of the peak stress at higher frequencies and the larger distribution of high

stress areas around the damage.

3.3.3: Results of Damage Case 3 with Center and Diagonal Loading

In damage case 3, the area of the damaged region in the corner of the sheet is

enlarged. For both the central and diagonal load cases, the stress at the center of the

damaged region, through each layer, is nearly identical to that of damage case 1. But,


43/58

37

when the sheet is viewed from the top, the stress in the area just below the damaged

region increases significantly as compared to the baseline, or damage cases 1 and 2.




44/58

38

The increase in stress, seen in Figures 3.33 and 3.34, is brought about by the shift

in the load path of the material in this area of the sheet. Since the damage is enlarged, the

area surrounding the damage has to accommodate the load even more than in cases 1 and

2. This becomes the dominant area in which the damage will develop further. As the

damage region continues to widen, the stress at the boundaries of the damage will

continue to rise, thus perpetuating the failure within the composite.

In terms of a frequency shift, due to the reduction in stiffness, modes 3 and 5 are

shifted down 1 Hz, and mode 4 is shifted down 1.5 Hz. As the damaged region grows

larger, the more influence it will have on other mode shapes.


In this case, the elements at the center of the sheet were modeled with damage.

The center load was applied right over the damaged region. But in the case of the

diagonal load, the damage was still applied at the center of the sheet, as opposed to

beneath the load. This allowed for the comparison of the stresses induced by direct

loading versus the stresses induced by an excited mode. Due to the extreme variation in

the results from the center load and diagonal load, one of the plots produced from the

diagonal load are presented and only two of the center load results are presented.

Figure 3.35: Layer 2 of Case 4 with Diagonal Loading, Stress vs. Frequency


45/58

39

Figure 3.36: Layer 1 of Case 4 with Central Loading, Stress vs. Frequency



46/58

40

The peak stresses occur at the same frequency of 37 Hz. The higher frequency

mode is not as prevalent when the damage is present in the center of the sheet. The

magnitude of stress in Figure 3.36 is the largest of any other data taken from the Ansys

modeling, with a value of about 210 MPa. For this case, the stress rises around the entire

damage region. As the load is applied directly to the damage, the load path is through all

layers and in all directions away from the damage. In this particular case, the composite

sheets will failure quicker with the load applied to the damage location. In other cases,

such as notches at the base of a cantilever, the part will fail quicker if the load is applied

farther away from the notch.


The damage modeled in cases 1 through 4 was based on the non-symmetric

failure of 10 consecutive layers, within a specified region. In case 5, the depth of the

damage was increased to 14 layers and was symmetric about the mid-plane of the sheet.

For this case, the stresses in the layers above and below the damage are essentially the

same, within symmetric layers (i.e. the first and last layers with the same orientation of

zero). The center-loaded case displayed a maximum stress in the first and twentieth layer

is about 69 MPa at 107 Hz, and the maximum stress in the second and nineteenth layer is

about 72 MPa at 37 Hz. In the diagonal-loaded case, the maximum stress in layers 1 and

20 was 65 MPa at 63 Hz, and the maximum stress in layer 2 and 19 was 50 MPa at 107

Hz. These magnitudes are higher than those found in the top layers of cases 1 and 2.

This is because the stress is distributed evenly in the reduced areas above and below the

damage. These areas act as equal load carrying paths.


47/58

41

3.4: Force Profile as Applied to Impact Damage

The characteristics and location of the forces applied to composite structures can

have a significant effect on the life of that structure. The relation of the impact time step

and the activated frequency band is simply f =1/dt, the shorter the time step, the broader

the frequency range that is excited. Figure 3.38 shows the time step for three different

unit impact loads, which are 8 ms, 48 ms, and 178 ms.

Figure 3.38: Force Profile of an Impact Load, Time and Frequency Based

The time steps used in Figure 3.38 produce initial frequency bands of 125 Hz, 20.8 Hz,

and 5.6 Hz, respectively. Each of frequency bands, that are associated with a given input

time step, represent the envelope in which energy is absorbed into the system. It should

be noted that at the anti-nodes in the frequency plot, there is not energy absorbed into the

system. Therefore, if a resonance falls on one of those frequencies, the excess energy


48/58

42

that is generated from that mode will not be absorbed by the system. Conversely, as an

example, the 8 ms impact would excite the first five modes of the modeled composite

sheet. Because each stress peak would be located within the first frequency band (shown

in blue), all of the energy from those modes would be absorbed into the system. It is

situations like this that must not occur in order to avoid the progression of damage in the

structure.

When damage accumulates in a structure the dynamic characteristics, such as

natural frequency, change with time and use. If the frequency of a mode that was once

outside of the absorption band decreases, the energy from that mode will then be

absorbed if the input remains the same. Reliability forecasting is based on lower

frequency phenomenon. In this project the response of the composite sheet changed with

respect to the input location and frequency. Because the inner layers of composites

absorb the energy from impacts, the damage was modeled therein. The model was set up

as a linear system; therefore the input force and stress at any location are related in this

manner. If the force input were to increase, the stress in the sheet would increase

proportionally to the input.

3.5: Discussion of Damage Accumulation, with Examples

As composite structures are exposed to numerous loads, such as ballistic impacts,

the composition of the material becomes damaged. From the analyses conducted, it can

be seen the severity of that damage can affect the outlying areas of the structure. The

location of the damage, as well as the location and duration of the loading, can ultimately

lead to the failure of the structure. In damage case 1, discussed in section 3.3.1, the stress

at the center of the damaged region was measured through the thickness as a function of

frequency. The stress field generated at the damaged location by the center load

condition is shown in Figures 3.39 through 3.41.


49/58

43

Figure 3.39: Stress Field Generated from the Center Load at the Damage Center

Figure 3.40: Stress Field from Center Load, Stress vs. Frequency


50/58

44

Figure 3.41: Stress Field from the Center Load, Layer Number vs. Frequency

The stress field measured through the thickness of the composite as a function of

frequency, due to the diagonal loading of damage case 1, is shown in Figures 3.42

through 3.44. Extracting the data from a specific node in the finite element model, and

then storing it in MATLAB generated the plots in this section. This data was then

processed to form the continuous stress field through the composite.


51/58

45

Figure 3.42: Stress Field Generated from the Diagonal Load at the Damage Center

Figure 3.43: Stress Field from Diagonal Load, Stress vs. Frequency


52/58

46

Figure 3.44: Stress Field from the Diagonal Load, Layer Number vs. Frequency

Each set of results are based on the magnitude of the real and imaginary part of

the Von Mises Equivalent stress response due to the load input on the sheet. The Von

Mises Equivalent Stress is represented here, but the principal stresses in each direction,

within each layer can also be calculated as well. The principal stress could be more

beneficial in that they could be compared to the tensile strength in the fiber direction and

transverse fiber direction of each layer. Heterogeneous materials, like composites, do not

have a single value for their yield or ultimate strength. Rather, their strength varies from

layer to layer.

In terms of damage accumulation, the damage that was modeled in each of these

cases was the failure of the ten consecutive plies in a square region of 4x4 elements. The

damaged area is indicated by the deep blue color in the center of the layer number vs.

frequency plots, which means there is no stress in this region. It can be seen from the

results that the frequency and layer of the peak stress shift when the load is applied to a

different location. As the different natural modes are excited, the progression of the


53/58

47

damage will take place in the layers that show the highest stresses. The damage case

with the load at the center of the plate has the largest overall magnitude of stress, as well

as the largest peak at the higher frequency mode. This can be a major concern in terms o

the fatigue that will take place from having a larger oscillating stress. Both have the

largest stresses in the in the top three layer of the composite. But in comparing the

results, the case that is more beneficial to the one being protected by the composite sheet

would be the impact at the location diagonal to the center. There are several reasons for

this:

1. The peak stress, overall, is not as large

2. The high frequency mode does not have the largest stress

3. The lowest mode is not as excited

The last reason is based on the load time step. The larger the time step of the impact, the

more both cases benefit. In the case where the load is applied diagonal to the center, the

first mode would not be excited by a 25 ms impact, but the center load case would.

The damage accumulates as failures persist within the matrix and the fiber, as the

maximum stress for that ply is exceeded the more failure will occur.


54/58

48

CHAPTER 4: SUMMARY, CONCLUSTIONS, AND RECOMMENDATIONS FOR

FUTURE RESEARCH

4.1: Summary and Conclusions

Chapter 1 outlined the benefits of utilizing heterogeneous materials, such as

composites, in a broad array of applications. It also specified the need for a deeper

understanding of the failure mechanisms in composites. If accurate predictions can be

made about the life expectancy of these materials, then reliability during use and in

maintenance will increase. The damage mechanisms that occur inside of composites can

be broken down into four major contributors; matrix cracking, delamination, fiber

breakage, and local buckling. These mechanisms interact with one another, thus

producing progressive damage accumulation.

Past research has been done to in both experimental and analytical testing of

various composite structures. Theses studies focused on the mathematical representation

of damage in composites and also in measure the progression of damage in composite

systems. Many of the analytical models used failure criteria such as Hashin or Hill-Tsai.

Chapter 2 focused on the techniques used to generate the finite element model in

Ansys, as well as the data used to represent the physical part. Much time was spentlearning how to create a realistic model using the tools within Ansys. The model was

based on a 20-ply laminated composite made of S2 Glass/Epoxy. The benefit of S glass

is that it is made of magnesium aluminosilicate and offers higher tensile strength than E

glass. Realistic loads were applied to simulate a ballistic load.

Chapter 3 presented the results of the damage cases and load conditions. There

were 5 damage cases in which the damage was re-located or re-sized. The loading

conditions were based on a distributed load that was either in the center of the composite

sheet of diagonal to the center. Numerous plots were generated in Ansys, to visually

show the distribution of stress within each composite layer. A consistent pattern was

seen in the damage cases in which the damage was not symmetric about the mid-plane.

In these cases the stress in the top layers, just above the damage, were higher than those


55/58

49

just below. This is attributed to the reduction in the local cross-sectional area. In some

cases the stress around the damage area would increase or decrease. The areas of

increase indicate the load-carrying path of the structure in that region. All of the damage

cases were compared to a “healthy” baseline.

Some of the data from the first damage case was stored in MATLAB, in order to

generate plots of the stress field through the thickness of the composite. This data was

then used to describe the progression and accumulation of damage in the composite.

Also, the force profile used as the input for loading a composite sheet was discussed in

terms of its effects on stress when applied at different time steps and at different

magnitudes.

In conclusion, by using stress as a measure of the load-carrying path, predictions

can be made about where the damage will grow. The effects of the force location and

size were found to have a significant impact on the response of the composite sheet. The

finite element model can be used as a means of evaluating the change in the stress

distribution due to damage in multiple locations within a composite sheet. Also, the

affects of applying the load in different locations provided valuable insight as to the

levels of stress in the composite at different frequencies. The load applied at the center of

the damage location generated the largest stress in each of the damage cases. This

excited the highest mode, of those looked at, which can lead to higher fatigue in the

composite

4.2: Recommendations of Future Work

The present research has demonstrated that FEA is a valuable tool for comparing

the affects of damage and loading location on the response of a composite laminate. But

there is still much work that can be pursued in this area, in terms of other loading and

boundary conditions. Also, there are issues that need further investigation.

• The use of the reduced stiffness properties to represent the damaged regions

should be verified with further research.

• The load can be applied in more locations or at different magnitudes.


56/58

50

• The frequency range of the harmonic forced response can be shifted to evaluate

higher modes.

• Transient analyses can be run to apply a true time step load to the composite; also

transient results can be generated into transfer functions for the composite.

• A different metric could be looked, instead of stress, as means of evaluating the

progression and accumulation of damage.


57/58

51

LIST OF REFERENCES

[1] Cantwell, W. J., Curtis, P. T., and Morton, J., “Impact and Subsequent FatigueDamage Growth in Carbon Fibre Laminates,” International Journal of Fatigue,Vol. 6,

No. 2, April 1984, pp.113-118.

[2] Craig, R. R., Mechanics of Materials, 2nd Edition New York, New York: John Wiley

& Sons 2000.

[3] Engblom, J. J., Havelka, J. J., “Combined Analytical/Experimental Approach for

Developing Structural Model of Damaged Composite Structures,” AIAA Conference,

American Institute of Aeronautics and Astronautics, AIAA-91-1085-CP, 1991.

[4] Hashin, Z., “Failure Criteria for Unidirectional Fiber Composites,” Journal of

Applied Mechanics, V 47, 1980, pp. 329-334.

[5] Hertzberg, R. W., Deformation and Fracture Mechanics of Engineering Materials,

4th Edition New York, New York: John Wiley & Sons 1996.

[6] Lee, J. D., “Three Dimensional Finite Element Analysis of Damage Accumulation in

Composite Laminates,” Computers and Stuctures, Vol. 15, No. 3, 1982, pp. 335-350.

[7] Reddy, A. D., Rehfield, L. W., and Haag, R. S., “Influence of PrescribedDelaminations on Stiffness-Controlled Behavior of Composite Laminates,” Effects of

Defects in Composite Materials, ASTM STP 836 , American Society for Testing and

Materials, 1984, pp. 71-83.

[8] Sun, C. T., Mechanics of Composite Materials and Laminates, Lecture Notes A&AE

555, Spring 2003, Purdue University.

[9] Talreja, R., “Stiffness Properties of Composite Laminates with Matrix Cracking and

Interior Delamination,” Engineering Fracture Mechanics, Vol. 25, No. 5/6, 1986, pp.751-762.

[10] Thompson, W. T., Dahleh, M. D., Theory of Vibration with Applications, 5th EditionUpper Saddle River, New Jersey: Prentice Hall 1998.

[11] Yen, C., Cassin, T., Patterson, J., Triplett, M., “Progressive Failure of Thin Walled

Composite Tubes Under Low Energy Impact,” U.S Army Missile Command, 1997.


58/58

52

Date post:	06-Jul-2018
Category:	Documents
Upload:	aman-tiwari
View:	217 times
Download:	0 times

Bsme Thesis Jfw

Documents