Modelling of Sandwich Composite Specimen with MFC actuators: Static Simulation on Bending.

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————————————————————— Civil and Industrial Engineering Department Master of Science in Aerospace Engineering

MODELLING OF SANDWICH COMPOSITE SPECIMEN WITH

MFC ACTUATORS: STATIC SIMULATION ON BENDING

Supervisor: Candidate: Prof. Mario Rosario Chiarelli Ashwik Manjunadh Sai Teja Nallagopulla

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Graduation Session of 2 May 2018 Academic year 2016-17

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Abstract

The aim of the thesis is to investigate and evaluate the static bending behaviour of sandwiched composite fibre specimen with piezoelectric patches (MFC) glued on to them, carried out with the parametric software ANSYS.

The composite material utilised are made of Glass/Epoxy or Graphite/Epoxy (Face sheets) prepreg fabric with 0.125 mm thickness. The piezoelectric patches, Macro Fibre Composite (MFC) are with thickness of 0.3 mm, which are the active control element of the structure. MFC patches are glued on to face sheets, and the whole specimen is then sandwiched with honeycomb as core. The chosen honeycomb is orthotropic material with thickness 3mm. Static simulation work for bending analysis is carried out under a voltage load applied on active area of MFC. Dimensions for the model are selected from bending composite specimen from Smart Material Corp. (SMC).

In the final phase of study, Glass/Epoxy & Graphite/Epoxy at the bottom face are replaced with a new specimen called PVC-P, a polymer. The complete study is focused on maximum vertical displacement that can be obtained when a voltage load is applied on active areas of MFC with consideration of honeycomb structure as core element.

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Acknowledgements

First, I would like to thank Prof. Mario Rosario Chiarelli for his support and for the direction he provided for my thesis work. The door to Prof. Chiarelli office was always open whenever I ran into trouble spot or had a question about my thesis work or writing. He consistently allowed this thesis to be my own work but steered me in the right direction whenever he thought I needed it. Without his guidance, none of this work would have been possible.

I would like to thank all my friends for their support, precisely, Mr. Bhargav & Mr. Aditya for their support in my hard times. I would like to thank “Associazione Malatesta Onlus” for their endless support.

Finally, words are not enough to express my gratitude to my parents and to my brother for providing me unfailing support and continuous encouragement throughout my years of study. This accomplishment of Master’s Degree Title would have not been possible without them. Thank you.

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Introduction

The idea of building lighter, more efficient and more aerodynamic aircraft is crucial for saving on energy and costs. In the recent decades, a lot of research has been carried out on integrating piezoelectric material, which can be used as actuators and sensors into base structure to control the structure.

The project was focused on the study of a futuristic wing having the capability of changing its aerodynamic shape (“self-shaping wing”) through the use of hybrid materials, made up of piezoelectric patches co-cured with a composite substrate.

For a morphing wing, traditional control surfaces (such as ailerons, flaps, slats and so on) are no longer required; that allows us to save weight in wing structures and reduce the sources of vibrations.

The deformed shape of a wing required by a given flight manoeuvre is obtained as a result of medium/high voltages applied to the active piezo-electric patches. This required a proper design of the hybrid active composite laminate.

In the project a very important amount of work has been dedicated to modelling and static structural simulation of the hybrid specimen. This hybrid specimen is a sandwiched model composed of carbon/epoxy as composite material (Face sheets) and honeycomb beam as core element of the sandwiched model. Macro Fibre Composite (piezoelectric patches) are glued on the top and bottom surface of carbon/epoxy composite material. The carbon/epoxy composite material is four layered face sheets with thickness 0.125 mm. The honeycomb core is one single structural beam with orthotropic material and thickness of 3mm. The Macro Fibre Composite is supplied by Smart material corp. The material is M8528-P1, with fibre of 0 degree for the bending specimen.

The majority of the work carried out was on numerical simulation of the structure with Finite Element Analysis and computer aided engineering software ANSYS to generate model, which provides the static bending behaviour of the hybrid structure. The static structural analysis provides us with the information of how the hybrid structure responds to voltage load and helps us to understand the maximum achievable vertical displacement of the structure (bending nature).

The first 2 chapters provides brief background on Piezoelectricity, MFC, Sandwich Structure & Composite fibres and the governing equation, followed by description of various samples utilized for the study with numerical simulation results and concluding with introduction to new specimen -Polymer. The complete study was focused on maximum vertical displacement (bending nature) that structure can obtain under voltage load, applied on active areas of the Macro Fibre Composite (piezoelectric actuator).

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Chapter 1

1.1 Composite Material

The word composite signifies that two or more materials are combined on a macroscopic scale to form a useful third material. The key is the macroscopic examination of a material wherein the components can be identified by the naked eye. Different materials can be combined on a microscopic scale, such as in alloying of metals, but the resulting material is, for all practical purposes, macroscopically homogenous, i.e., the components cannot be distinguished by the naked eye and essentially act together. The advantage of composite materials is that, if well designed, they usually exhibit the best qualities of their components or constituents. Some of the properties that can be improved by forming a composite material are

• Strength • Stiffness

• Corrosion resistance • Weight

• Fatigue life

Naturally, not all of these properties are improved at the same time nor is there usually any requirement to do so. The objective is merely to create a material that has only the characteristics needed to perform the design task.

Composites materials have long history of usage. Their precise beginnings are unknown, but all the recorded history contains references to some form of composite material. For example, Plywood was used by the ancient Egyptians when they realized that wood could be rearranged to achieve superior strength and resistance to thermal expansion as well as to swelling caused by the absorption of moisture. More recently, fibber-reinforced, resin-matrix composite materials that have high strength-to-weight ratios have become important in weight sensitive applications such as aircraft and space vehicles.

1.1.1 Classification and Characteristics of Composite Materials

Four commonly accepted types of composite materials are:

1. Fibrous composite materials that consist of fibres in a matrix

2. Laminated composite materials that consist of layers of various materials 3. Particular composite materials that are composed of particles in a matric 4. Combinations of some or all of the first three types

1.1.1.a Fibrous Composite Materials

Long fibbers in various forms are inherently much stiffer and stronger than same material in bulk form. For example, ordinary plate glass fractures at stresses of only a few thousand pounds per square inch (lb/in2 or psi) (20 MPa), yet glass fibbers have strengths of 400,000 to 700,00 psi (2800 to 4800 MPa) in commercially available forms and about 1,000,000 psi (7000 MPa) in laboratory-prepared forms. Obviously, then, the geometry ad physical makeup of a fibre is somehow crucial to the evaluation of its strength and must be

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considered in structural applications. More properly, the paradox of a fibre having different properties from the bulk form is due to the more perfect structure of a fibre. In fibers, the crystals are aligned along the fiber axis. Moreover, there are fewer internal defects in fibres than in bulk material. For example, in materials that have dislocations, the fibre form has fewer dislocations than bulk form. A fibre is characterized by geometrically not only by its very high length-to-diameter ratio but by its near-crystal-sized length-to-diameter.

Graphite or carbon fibers are of high interest in today’s composite structures. Both are made from rayon, pitch, or PAN (polyacrylonitrile) precursor fibres that are heated in an inert atmosphere to about 3100 °F to carbonize the fibres. To get graphite fibres, the heating exceeds 3100°f to partially graphitize the carbon fibres. Actual processing is proprietary, but fibre tension is known to be a key processing parameter. Moreover, as the processing temperature is increased, the fibre modulus increases, but the strength often decreases. The fibres re actually far thinner than human hairs, so they can be bent quite easily. Thus, carbon or graphite fibres can be woven into fabric.

1.1.2 Mechanical Behaviour of Composite Materials

Composite materials have many mechanical behavior characteristics that are different from those of more conventional engineering materials. Some characteristics are merely modifications of conventional behavior; others are totally new and require new analytical and experimental procedures.

Most common engineering materials are both homogeneous and isotropic:

A homogeneous body has uniform properties throughout, i.e., the properties are independent of the position in the body.

An isotropic body has material properties that are the same in every direction at a point in the body, i.e., the properties are independent of orientation at a point in the body. Bodies with temperature-dependent isotropic material properties are not homogeneous when subjected to a temperature gradient, but still are isotropic.

In contrast, composite materials are often both inhomogeneous (or nonhomogeneous or heterogeneous – the three terms can be used interchangeably) and nonisotropic (orthotropic or, more generally, anisotropic, but the word are not interchangeable):

An inhomogeneous body has non-uniform properties over the body, i.e., the properties depend on position in the body.

An orthotropic body has material properties that are different in three mutually perpendicular directions at a point in the body and, further, has three mutually perpendicular planes of material property symmetry. Thus, the properties depend on orientation at a point in the body.

An anisotropic body has material properties that are different in all directions at a point in the body. No planes of material property symmetry exist. Again, The properties depend on orientation at a point in the body.

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inhomogeneity of the composite material is a step function in the direction perpendicular to the plane of the glass. Also, some particulate composite materials are inhomogeneous, yet isotropic, although some are orthotropic, and others are anisotropic. Other composite materials are typically more complex, especially those with fibres places=d at many angles in space.

Because of the inherently heterogeneous nature of composite materials, they are conveniently studied from two points of view: micromechanics and macromechanics:

Micromechanics is the study of composite material behavior wherein the interaction of the constituent materials is examined on a microscopic scale to determine their effect on the properties of the composite material

Macromechanics is the study of composite material behavior wherein the material is presumed homogeneous and the effects of the constituent materials are detected only as apparent macroscopic properties of the composite material.

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1.1.3 Why Fibre Orientation?

The strength and stiffness of a composite buildup depends on the orientation sequence of the plies. The practical range of strength and stiffness of carbon fiber extends from values as low as those provided by fiberglass to as high as those provided by titanium. This range of values is determined by the orientation of the plies to the applied load. Proper selection of ply orientation in advanced composite materials is necessary to provide a structurally efficient design. The part might require 0° plies to react to axial loads, ±45° plies to react to shear loads, and 90° plies to react to side loads. Because the strength design requirements are a function of the applied load direction, ply orientation and ply sequence have to be correct. It is critical during a repair to replace each damaged ply with a ply of the same material and ply orientation.

Figure 2 Quasi-isotropic material lay-up.

1.1.4 Types of Fiber

a) Fiberglass

Fiberglass is often used for secondary structure on aircraft, such as fairings, radomes, and wingtips. Fiberglass is also used for helicopter rotor blades. There are several types of fiberglass used in the aviation industry. Electrical glass, or E-glass, is identified as such for electrical applications. It has high resistance to current flow. E-glass is made from borosilicate glass. S-glass and S2-glass identify structural fiberglass that have a higher strength than E-glass. S-glass is produced from magnesia-alumina-silicate. Advantages of fiberglass are lower cost than other composite materials, chemical or galvanic corrosion resistance, and electrical properties (fiberglass does not conduct electricity). Fiberglass has a white color and is available as a dry fiber fabric or prepreg material.

b) Kevlar®

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resistance to impact damage, so they are often used in areas prone to impact damage. The main disadvantage of aramid fibers is their general weakness in compression and hygroscopy. Service reports have indicated that some parts made from Kevlar® absorb up to 8 percent of their weight in water. Therefore, parts made from aramid fibers need to be protected from the environment. Another disadvantage is that Kevlar® is difficult to drill and cut. The fibers fuzz easily and special scissors are needed to cut the material. Kevlar® is often used for military ballistic and body armor applications. It has a natural yellow color and is available as dry fabric and prepreg material. Bundles of aramid fibers are not sized by the number of fibers like carbon or fiberglass but by the weight.

c) Carbon/Graphite

One of the first distinctions to be made among fibers is the difference between carbon and graphite fibers, although the terms are frequently used interchangeably. Carbon and graphite fibers are based on graphene (hexagonal) layer networks present in carbon. If the graphene layers, or planes, are stacked with three-dimensional order, the material is defined as graphite. Usually extended time and temperature processing is required to form this order, making graphite fibers more expensive. Bonding between planes is weak. Disorder frequently occurs such that only two-dimensional ordering within the layers is present. This material is defined as carbon.

Carbon fibers are very stiff and strong, 3 to 10 times stiffer than glass fibers. Carbon fiber is used for structural aircraft applications, such as floor beams, stabilizers, flight controls, and primary fuselage and wing structure. Advantages include its high strength and corrosion resistance. Disadvantages include lower conductivity than aluminum; therefore, a lightning protection mesh or coating is necessary for aircraft parts that are prone to lightning strikes. Another disadvantage of carbon fiber is its high cost. Carbon fiber is gray or black in color and is available as dry fabric and prepreg material. Carbon fibers have a high potential for causing galvanic corrosion when used with metallic fasteners and structures.

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Figure 3 Fiberglass (left), Kevlar® (middle), and carbon fiber material (right).

1.1.5 Matrix Materials

a) Epoxy

Epoxies are polymerizable thermosetting resins and are available in a variety of viscosities from liquid to solid Epoxies are used widely in resins for prepreg materials and structural adhesives. The advantages of epoxies are high strength and modulus, low levels of volatiles, excellent adhesion, low shrinkage, good chemical resistance, and ease of processing. Their major disadvantages are brittleness and the reduction of properties in the presence of moisture. The processing or curing of epoxies is slower than polyester resins. Processing techniques include autoclave molding, filament winding, press molding, vacuum bag molding, resin transfer molding, and pultrusion. Curing temperatures vary from room temperature to approximately 350 °F (180 °C). The most common cure temperatures range between 250° and 350 °F (120–180 °C).

c

1.1.6 Pre-Impregnated Products (Prepregs)

Prepreg material consists of a combination of a matrix and fiber reinforcement. It is available in unidirectional form (one direction of reinforcement) and fabric form (several directions of reinforcement).Prepreg materials are cured with an elevated temperature. Many prepreg materials used in aerospace are impregnated with an epoxy resin.

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1.2 Sandwich Structure

A sandwich construction is a structural panel concept that consists in its simplest form of two relatively thin, parallel face sheets bonded to and separated by a relatively thick, lightweight core. The core supports the face sheets against buckling and resists out-of-plane shear loads. The core must have high shear strength and compression stiffness. Composite sandwich construction is most often fabricated using autoclave cure, press cure, or vacuum bag cure. Skin laminates may be procured and subsequently bonded to core, co-cured to core in one operation, or a combination of the two methods. Examples of honeycomb structure are: wing spoilers, fairings, ailerons, flaps, nacelles, floor boards, and rudders.

Figure 4 Honeycomb sandwich construction.

1.2.1 Properties

Sandwich construction has high bending stiffness at minimal weight in comparison to aluminum and composite laminate construction. Most honeycombs are anisotropic; that is, properties are directional. Figure 5 illustrates the advantages of using a honeycomb construction. Increasing the core thickness greatly increases the stiffness of the honeycomb construction, while the weight increase is minimal. Due to the high stiffness of a honeycomb construction, it is not necessary to use external stiffeners, such as stringers and frames.

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Figure 5 Strength and stiffness of honeycomb sandwich material compared to a solid laminate.

1.2.2 Facing Materials

Most honeycomb structures used in aircraft construction have aluminum, fiberglass, Kevlar®, or carbon fiber face sheets. Carbon fiber face sheets cannot be used with aluminum honeycomb core material because it causes the aluminum to corrode. Titanium and steel are used for specialty applications in high temperature constructions. The face sheets of many components, such as spoilers and flight controls, are very thin—sometimes only 3 or 4 plies.

1.2.3 Core Materials

Each honeycomb material provides certain properties and has specific benefits. The most common core material used for aircraft honeycomb structures is aramid paper (Nomex® or Korex®). Fiberglass is used for higher strength applications.

a) Kraft paper—relatively low strength, good insulating properties, is available in large quantities, and has a low cost.

b) Thermoplastics—good insulating properties, good energy absorption and/or redirection, smooth cell walls, moisture and chemical resistance, are

environmentally compatible, aesthetically pleasing, and have a relatively low cost.  

c) Aluminum—best strength-to-weight ratio and energy absorption, has good heat transfer properties, electromagnetic shielding properties, has smooth, thin cell walls, is machinable, and has a relatively low cost  

d) Steel—good heat transfer properties, electromagnetic shielding properties, and heat resistant  

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f) Aramid paper—flame resistant, fire retardant, good insulating properties, low dielectric properties, and good formability.  

g) Fiberglass—tailorable shear properties by layup, low dielectric properties, good insulating properties, and good formability.  

h) Carbon—good dimensional stability and retention, high-temperature property retention, high stiffness, very low coefficient of thermal expansion, tailorable thermal conductivity, relatively high shear modulus, and very expensive  

i) Ceramics—heat resistant to very high temperatures, good insulating properties, is available in very small cell sizes, and very expensive.  

Honeycomb core cells for aerospace applications are usually hexagonal. The cells are made by bonding stacked sheets at special locations. The stacked sheets are expanded to form hexagons. The direction parallel to the sheets is called ribbon direction.  Bisected hexagonal core has another sheet of material cutting across each hexagon. Bisected hexagonal honeycomb is stiffer and stronger than hexagonal core. Over expanded core is made by expanding the sheets more than is needed to make hexagons. The cells of over expanded core are rectangular. Over expanded core is flexible perpendicular to the ribbon direction and is used in panels with simple curves. Bell- shaped core, or flexi core, has curved cell walls, that make it flexible in all directions. Bell-shaped core is used in panels with complex curves.  Honeycomb core is available with different cell sizes. Small sizes provide better support for sandwich face sheets. Honeycomb is also available in different densities. Higher density core is stronger and stiffer than lower density core.

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Figure 6 c)

Figure 6 Figures of 6 network describes Honeycomb density.

1.3 Piezoelectric Materials

Piezoelectricity is the ability of certain materials to become electrically charged if they are deformed or mechanically stressed (direct piezoelectric effect). In the same way, if these materials are crossed by an electric current or exposed to an electric field, they undergo to a deformation (inverse piezoelectric effect). Piezoelectric materials are therefore used to convert electrical energy into mechanical energy (such as actuators) and vice versa (such as sensors). The most common piezoelectric material is quartz. Certain ceramics, Rochelle salts, and various other solids also exhibit this effect. The behaviour of the piezoelectric materials has been analysed through the document. They are multi-functional material, which can be used for adaptable structural systems. They can be distinguished from the type of excitation mechanism, from there field of application and the status of development. In this section the piezoelectric effect and its constituent elements will be discussed.

1.3.1 Historical Background

The pyroelectric effect by which a material generates an electric potential in response to a variation of temperature, was studied by Carl Linnaeus and Franz Aepinus in the mid-18th century. Starting from these studies, both René Just Hauy and Antoine César Becquerel examined the relationship between the mechanical stress and the electric charge. The first demonstration of the direct piezoelectric effect was processed in 1880 by the brothers Pierre and Jaques Curie. They combined their knowledge of pyroelectricity with the understanding of the underlying crystal structures to predict crystal behaviour and they could demonstrate its effect using quartz, topaz, cane sugar and Rochelle salt. The Curies, however, did not predict the inverse piezoelectric effect. The latter was mathematically deduced by fundamental thermodynamic principles by Gabriel Lippmannin in 1881. The Curies immediately confirmed the existence of inverse effect and went to obtain quantitative proof of

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crystals. In 1910 with the publication of the work done by Woldemar Voigt: Lehrbuch der Kristqallphysik (Textbook on Crystal Phisics) were described the 20 natural crystal classes capable of piezoelectricity. He rigorously defined the piezoelectric elements using tensor analysis. The first practical application for piezoelectric devices was the sonar, developed during the First World War. In France in 1917 Paul Langevin and his coworkers developed an ultrasonic submarine detector. The detector consisted of a transducer, made of thin quartz crystals carefully glued between two steel plates, and a hydrophone to detect the returned echo. By emitting a high-frequency pulse from the transducer and measuring the amount of time it takes to hear the echo from sound waves bouncing on an object, the distance to that object can be calculated. Over the next few decades, new piezoelectric materials and new applications were explored and developed: ultrasonic transducers allowed easy measurements of viscosity and elasticity in fluids and solids; ultrasonic time-domain refractometers (which send an ultrasonic pulse through a material and measure reflections from discontinuities) could find flaws inside cast metal and stone objects, improving structural safety. During Second World War, independent research groups in USA, Russia and Japan discovered a new class of synthetic materials called ferroelectrics, which exhibited piezoelectric constants many times higher than the natural ones.

1.3.2 Crystalline Structure

Piezoelectric effect arises from a crystal structure. The piezoelectric effect is exhibited by 20 out of 32 crystal classes and is always associated with noncentrosymmetric crystals. Naturally occurring materials, such as quartz, exhibit this effect as a result of their crystalline structure. Engineered materials, like lead zirconate titanate (PZT) for instance, are subjected to a process called poling to impart the piezoelectric behavior.

Figure 7 A perovskite unit cell showing the, off centered titanium ion.

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titanium ion sitting slightly off-center inside the unit cell, an electrical polarity develops, thereby turning the unit cell effectively into an electric dipole. A mechanical stress on the crystal further shifts the position of the titanium ion, thus changing the polarization strength of the crystal. This is the source of the direct effect. When the crystal is subjected to an electric field, it also results in a relative shift in the position of the titanium ion, leading to the distortion of the unit cell and making it more (or less) tetragonal. This is the source of the inverse effect.

1.3.3 Polarization

Piezoelectric properties are due to crystalline structure of the materials. In a macroscopic crystalline structure that comprises several such unit cells, the dipoles are by default found to be randomly oriented. When the material is subjected to a mechanical stress, each dipole rotates from its original orientation toward a direction that minimizes the overall electrical and mechanical energy stored in the dipole. If all the dipoles are initially randomly oriented (i.e. a net polarization of zero), their rotation may not significantly change the macroscopic net polarization of the material, hence the piezoelectric effect exhibited will be negligible. Therefore, it is important to create an initial state in the material such that most dipoles will be more-or-less oriented in the same direction. Such an initial state can be imparted to the material by poling it. The direction along which the dipoles align is known as the poling direction.

During poling, the material is subjected to a very high electric field that orients all the dipoles in the direction of the field. Upon switching off the electric field, most dipoles do not return back to their original orientation as a result of the pinning effect produced by microscopic defects in the crystalline lattice. This gives us a material comprising numerous microscopic dipoles that are roughly oriented in the same direction. It is noteworthy that the material can be de-poled if it is subjected to a very high electric field-oriented opposite to the poling direction or is exposed to a temperature higher than the Curie temperature of the material.

1.4 Macro Fiber Composite

The Macro Fiber Composite (MFC) was invented by NASA in 1996[]. Smart Material started commercializing the MFC as the licensed manufacturer and distributor of the patented invention worldwide in 2002. The MFC consists of rectangular piezo-ceramic rods sandwiched between layers of adhesive, electrodes and polymide film. The electrodes are attached to the film in an interdigitated pattern which transfers the applied voltage directly to and from the ribbon shaped rods. As a thin surface, it can be applied (normally bonded) to various types of structures or embedded in a composite structure. If a voltage is applied, it will bend or distort the material, counteract vibrations or generate them. If no voltage is applied, it can work as a very sensitive strain gauge, sensing deformations, noise and vibrations. The MFC is also an excellent device to harvest energy from vibrations. In our applications P1 type MFC for the bending specimens have been used. The P1 exploit the d33 effect for actuation and

elongate up to 1800 ppm f they operate at maximum allowed voltage- 500V, 1500V. The schematics of the MFC are shown in the figure:

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Figure 8 MFC showing on top and bending in bottom.

1.4.1 Application of Macro Fiber Composites

Macro Fiber Composite offers high performance, flexibility, reliability and low cost. The fact that it can be bonded as thin surface-conformable sheet to several structures makes it easy to be used. On application of electric excitation or voltage it will bend or distort material, counteract vibration or generate vibration. It can also be used as a strain gauge to sense vibration, noise and to harvest energy from the vibrations. In the recent years the Macro fiber composites have found applicability in numerous fields ranging from Aerospace/civil structures to Sonar application in submarines. Some of the prominent application include

1.4.1.1 Deformation of wing using MFC

Macro Fiber composites were used on the upper and lower surface of the wing to change its shape. The geometry of the wing is similar to NACA0014. The Macro Fiber Composites are bonded to the inside and become an integral part of the wing surface. Various wind tunnel experimental were carried in still-air and various flow regimes to understand the effect the change of shape of airfoil.

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Figure 9 Airfoil with MFC patch on the inner surface (left) and MFC patch on the (right).

1.4.1.2 Energy Harvester for Unmanned Aerial Vehicle

The piezoelectric patch applied onto the surface of unmanned aerial vehicle is used to generate energy from vibration. The energy generated by the use of piezoelectric patch and that by photovoltaic cell were compare. It can be concluded that both the piezoelectric and solar energy harvesting devices have the capability of charging energy storage devices. Preliminary tests show that during a 13- minute flight, the solar panels could charge a 170 mAh battery to 14% capacity and the piezoelectric patches could charge the EH300 4.6 mJ internal capacitor to 70% capacity.

Figure 10 MFC Patch on the UAV Wing.

1.5 Constitutive Equations

The piezoelectric effect connects a field of mechanical stress to an electrostatic one; the link between stress and its deformation, by the mechanical part, and the magnetic flow density combined with electric field, from the electrostatic part, gives the constitutive equations. In the following passage has been analyzed a very important homogeneous macroscopic material, whose properties are considered in the studies of T. H. BrockMann.

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1.5.1 Mechanical Fields

This work will be limited to the material with a connection between the local stress and the deformation, but the deformation history and the time dependence are not taken into account. So the removal of the loads leads to a complete reversal of the situation. The below work for the formulation and verification is been inspired from the studies of R Jones, Mechanics of Composite Materials.

Hooke’s law is the principle of physics that states that the force (F) needed to extend or compress a spring by some distance, scales linearly with respect to that distance.

F = KX

The stresses and strains of the material inside a elastic material are connected by a linear relationship that is mathematically similar to Hooke’s spring law, and is often referred to by that name.

However, the strain state in a solid medium around some point cannot be described by a single vector. The concerned material no matter how small, can be compressed, stretched, and sheared at the same time along different directions. It tells that the stresses in that material can be at once pushing, pulling, and shearing.

In order to capture this complex behavior, state of the medium around a point must be represented by two-second-order tensors, the strain tensor 𝜀𝜀 ( 𝑖𝑖𝑖𝑖 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑖𝑖𝑡𝑡𝑠𝑠𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓𝑓𝑓 𝑑𝑑𝑖𝑖𝑠𝑠𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑡𝑡𝑑𝑑𝑡𝑡𝑖𝑖𝑡𝑡) and the stress tensor 𝜎𝜎 (𝑓𝑓𝑡𝑡𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑟𝑟 𝑡𝑡ℎ𝑡𝑡 𝑓𝑓𝑡𝑡𝑠𝑠𝑡𝑡𝑓𝑓𝑓𝑓𝑖𝑖𝑖𝑖𝑟𝑟 𝑓𝑓𝑓𝑓𝑓𝑓𝑑𝑑𝑡𝑡 𝐹𝐹). Then the Hooke’s spring law for the elastic material

is given by

𝜎𝜎 = −c 𝜀𝜀 Where c is a fourth-order tensor, called as Stiffness tensor.

Assumption 1: the mechanical behavior of the materials will be assumed as linear and flexible. The Cauchy stress tensor [T] and the Green strain tensor [S] are of the second order and it can be connected to a linear elastic anisotropic material with a fourth order tensor. The 81 elastic constants of this tensor are reduced to 36 one, thanks to the symmetry of the stress and the deformation tensor, represented in a quadratic matrix 6×6 [c]. Thanks to the consideration on the potential of the elastic materials, this matrix is symmetrical, and the number of the independent elastic components are reduced to 36 to 21. using the idea of infinitesimal displacement, the constitutive relation follows this form according to the Hook’s law:

[T] = [c]. [S]; [S] = [s]. [T] (1.1)

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[T] = ⎣ ⎢ ⎢ ⎢ ⎢ ⎡𝜎𝜎_𝜎𝜎1₂ 𝜎𝜎3 𝜏𝜏12 𝜏𝜏23 𝜏𝜏31⎦ ⎥ ⎥ ⎥ ⎥ ⎤ [S]= ⎣ ⎢ ⎢ ⎢ ⎢ ⎡_𝜖𝜖𝜖𝜖1₂ 𝜖𝜖3 𝛾𝛾12 𝛾𝛾23 𝛾𝛾31⎦ ⎥ ⎥ ⎥ ⎥ ⎤ (1.2)

The composite materials with a regular distribution of the constituents along the main axes are an example of a material with 3 orthogonal planes of symmetry. The description of such properties requires 9 independent elastic constants:

⎣ ⎢ ⎢ ⎢ ⎢ ⎡_𝜎𝜎𝜎𝜎1₂ 𝜎𝜎3 𝜏𝜏12 𝜏𝜏23 𝜏𝜏31⎦ ⎥ ⎥ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎢ ⎢ ⎡𝐶𝐶_𝐶𝐶11 𝐶𝐶12 𝐶𝐶13 21 𝐶𝐶22 𝐶𝐶23 𝐶𝐶31 0 0 0 𝐶𝐶32 0 0 0 𝐶𝐶33 0 0 0 0 0 0 0 0 0 0 𝐶𝐶44 0 0 0 0 𝐶𝐶55 0 0 0 0 𝐶𝐶66⎦ ⎥ ⎥ ⎥ ⎥ ⎤ ⎣ ⎢ ⎢ ⎢ ⎢ ⎡𝜖𝜖_𝜖𝜖1₂ 𝜖𝜖3 𝛾𝛾12 𝛾𝛾23 𝛾𝛾31⎦ ⎥ ⎥ ⎥ ⎥ ⎤ (1.3) 𝐶𝐶11 =1 − 𝜈𝜈_Ε 23𝜈𝜈32 2Ε3∆ 𝐶𝐶22= 1 − 𝜈𝜈13𝜈𝜈31 Ε1Ε3∆ 𝐶𝐶12= 𝜈𝜈21_Ε+ 𝜈𝜈31𝜈𝜈23 2Ε3∆ 𝐶𝐶23 = 𝜈𝜈32+ 𝜈𝜈12𝜈𝜈31 Ε1Ε3∆ (1.4) 𝐶𝐶13 =𝜈𝜈31_Ε+ 𝜈𝜈21𝜈𝜈32 2Ε3∆ 𝐶𝐶33= 1 − 𝜈𝜈12𝜈𝜈21 Ε1Ε2∆ 𝐶𝐶44 = 𝐺𝐺23 𝐶𝐶55= 𝐺𝐺31 𝐶𝐶66= 𝐺𝐺12 Where, ∆ = 1 − 𝜈𝜈12𝜈𝜈21− 𝜈𝜈23𝜈𝜈32− 𝜈𝜈31𝜈𝜈13− 2𝜈𝜈21𝜈𝜈32𝜈𝜈13 Ε1Ε2Ε3 Here to verify, 𝜈𝜈21𝜈𝜈32𝜈𝜈13 < 1 − 𝜈𝜈212 Ε_Ε1₂− 𝜈𝜈322 _ΕΕ2₃− 𝜈𝜈132 Ε_Ε3₁ 2 < 1 2 (1.5)

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− �𝜈𝜈32𝜈𝜈13Ε_Ε2 1+ �1 − 𝜈𝜈32 2 Ε2 Ε3 �1 − 𝜈𝜈13 2 Ε3 Ε1� Ε2 Ε1� < 𝜈𝜈21 < − �𝜈𝜈32𝜈𝜈13Ε_Ε2 1 − �1 − 𝜈𝜈32 2 Ε2 Ε3 �1 − 𝜈𝜈13 2 Ε3 Ε1� Ε2 Ε1� And, Ε1, Ε2, Ε3 = 𝑌𝑌𝑓𝑓𝑠𝑠𝑖𝑖𝑟𝑟′𝑠𝑠 𝑑𝑑𝑓𝑓𝑑𝑑𝑠𝑠𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖 𝑡𝑡ℎ𝑡𝑡 1 − ,2 − ,3 − 𝑑𝑑𝑖𝑖𝑓𝑓𝑡𝑡𝑑𝑑𝑡𝑡𝑖𝑖𝑓𝑓𝑖𝑖𝑠𝑠 𝜈𝜈𝐼𝐼𝐼𝐼 = 𝑃𝑃𝑓𝑓𝑖𝑖𝑠𝑠𝑠𝑠𝑓𝑓𝑖𝑖′𝑠𝑠 𝑓𝑓𝑑𝑑𝑡𝑡𝑖𝑖𝑓𝑓 𝑖𝑖. 𝑡𝑡., (𝑓𝑓𝑑𝑑𝑡𝑡𝑖𝑖𝑓𝑓 𝑓𝑓𝑓𝑓 𝑡𝑡ℎ𝑡𝑡 𝑡𝑡𝑓𝑓𝑑𝑑𝑖𝑖𝑠𝑠𝑡𝑡𝑡𝑡𝑓𝑓𝑠𝑠𝑡𝑡 𝑠𝑠𝑡𝑡𝑓𝑓𝑑𝑑𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑗𝑗 − 𝑑𝑑𝑖𝑖𝑓𝑓𝑡𝑡𝑑𝑑𝑡𝑡𝑖𝑖𝑓𝑓𝑖𝑖 𝑓𝑓𝑡𝑡𝑡𝑡𝑓𝑓 𝑡𝑡ℎ𝑡𝑡 𝑠𝑠𝑡𝑡𝑓𝑓𝑑𝑑𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑡𝑡ℎ𝑡𝑡 𝑖𝑖 − 𝑑𝑑𝑖𝑖𝑓𝑓𝑡𝑡𝑑𝑑𝑡𝑡𝑖𝑖𝑓𝑓𝑖𝑖 𝑤𝑤ℎ𝑡𝑡𝑖𝑖 𝑠𝑠𝑡𝑡𝑓𝑓𝑡𝑡𝑠𝑠𝑠𝑠 𝑖𝑖𝑠𝑠 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑖𝑖𝑡𝑡𝑑𝑑 𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡ℎ 𝑖𝑖 − 𝑑𝑑𝑖𝑖𝑓𝑓𝑡𝑡𝑑𝑑𝑡𝑡𝑖𝑖𝑓𝑓𝑖𝑖) 𝐺𝐺23, 𝐺𝐺31, 𝐺𝐺12 = 𝑠𝑠ℎ𝑡𝑡𝑑𝑑𝑓𝑓 𝑑𝑑𝑓𝑓𝑑𝑑𝑠𝑠𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖 𝑡𝑡ℎ𝑡𝑡 2 − 3, 3 − 1, 𝑑𝑑𝑖𝑖𝑑𝑑 1 − 2 𝑑𝑑𝑑𝑑𝑑𝑑𝑖𝑖𝑡𝑡𝑠𝑠 𝐶𝐶𝐼𝐼𝐼𝐼− 𝑠𝑠𝑡𝑡𝑖𝑖𝑓𝑓𝑓𝑓𝑖𝑖𝑡𝑡𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑𝑡𝑡𝑓𝑓𝑖𝑖𝑚𝑚 𝑑𝑑𝑖𝑖𝑑𝑑 𝑠𝑠𝐼𝐼𝐼𝐼− 𝑑𝑑𝑓𝑓𝑑𝑑𝑑𝑑𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑓𝑓𝑖𝑖𝑚𝑚

These materials have a uniform distribution and unidirectional of the fibers aligned with the principal axes and it possesses in addition an isotropy plane in the transverse direction, whose behavior of the material is invariant of the rotation. For this material the number of elastic constants is reduced to 5. When the rotation axes is oriented on the 𝑡𝑡₃ direction we have:

𝐶𝐶₂₂= 𝐶𝐶₁₁, 𝐶𝐶₂₃= 𝐶𝐶₁₃, 𝐶𝐶₄₄= 𝐶𝐶₅₅, 𝐶𝐶₆₆ = 𝐶𝐶11−𝐶𝐶12

2 (1.6)

If in any arbitrary plane section, there is an isotropy plane, the material is isotropic, so it remains only two elastic independent constants

𝐶𝐶₃₃ = 𝐶𝐶₂₂= 𝐶𝐶₁₁, 𝐶𝐶₂₃= 𝐶𝐶₁₃ = 𝐶𝐶₁₂, 𝐶𝐶₄₄ = 𝐶𝐶₅₅= 𝐶𝐶₆₆= 𝐶𝐶11−𝐶𝐶12

2 (1.7)

1.5.2 Electrostatic fields

Assumption 2: the electrostatic behaviour of the analysed material taken as linear. The electric flow density D and the electric field are vectors, namely first order tensor so it can be linked by a second order tensor with 9 elastic constants for the three dimensions. The properties observed in the electrostatic field, tell us that the tensor is symmetric, and it contains 6 independent parameters. The electrostatic relation can be expressed with the help of the electric permittivity matric [𝜖𝜖] and [𝜖𝜖₀], indicates the vacuum dielectric constant:

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D = 𝜖𝜖 . 𝐸𝐸 ; 𝜖𝜖 = 1

𝑐𝑐2 _𝜇𝜇₀ ~ 8.854187816 . 10−12 𝐹𝐹_𝑚𝑚; (1.8) 𝜖𝜖 = 𝜖𝜖̅. 𝜖𝜖0

The crystal that possesses rhombic crystalline matrix as a composite material, with adequate provision behave as an orthotropic material and it presents only three parameters along the diagonal of the constituent matrix:

�𝐷𝐷𝐷𝐷12 𝐷𝐷3 � = �𝜖𝜖011 𝜖𝜖022 00 0 0 𝜖𝜖33 � �𝐸𝐸𝐸𝐸12 𝐸𝐸3 � (1.9)

Isotropic properties are exhibited transversely, for example, by the tetragonal crystal structures which require only two dielectric constants:

𝑡𝑡11= 𝑡𝑡22 (1.10)

Isotropic properties are exhibited in the cubic crystalline structures, where there is only one dielectric constant:

𝑡𝑡11= 𝑡𝑡22 = 𝑡𝑡33 (1.11)

1.5.3 Electromechanical Coupling

Consider the assumptions 1 and 2, also the coupling between the mechanical and electrostatic fields will be imitated only at a linear case if we consider the piezoelectricity and neglect the nonlinear effect.

Assumption 3: electro-mechanical coupling in this material will be assumed linear. The piezoelectricity is described inside the constitutive equation of the material, which defines how the stress of the piezoelectric material, the displacement field, electric charge density and the electric field interact. So, the piezoelectric material combines the two constitutive equations of the assumption 1 and 2, apparently dissimilar consequently, the coupling equation is written as follows:

𝑆𝑆 = 𝑠𝑠𝑒𝑒. 𝑇𝑇 + 𝑑𝑑𝑡𝑡. Ε;

(1.12) 𝐷𝐷 = 𝑑𝑑. 𝑇𝑇 + 𝜖𝜖 . Ε;

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The d matrix contains the piezoelectric coefficient and it appears twice on the constitutive equation, (𝑑𝑑𝑡𝑡 𝑖𝑖𝑠𝑠 𝑡𝑡𝑓𝑓𝑑𝑑𝑖𝑖𝑠𝑠𝑑𝑑𝑓𝑓𝑠𝑠𝑡𝑡𝑑𝑑 𝑑𝑑𝑑𝑑𝑡𝑡𝑓𝑓𝑖𝑖𝑚𝑚). 𝑑𝑑 = � 00 00 00 𝑑𝑑31 𝑑𝑑32 𝑑𝑑33 𝑑𝑑015 𝑑𝑑015 00 0 0 0� (1.13)

In order to describe the piezoelectric material, the user must have a very good knowledge of the mechanical properties (compliance or elasticity), the electrical properties (permittivity) and the piezoelectric ones are known. The indices of the constitutive equations have a very important meaning. They describe the conditions whose properties of the material has been measured. The four state variables (S, T, D and E) can be rewritten and they give 3 additional modules for a constitutive piezoelectric equation. These variables contain the coupling matrix e, g, or q. So, it is possible to transform the constitutive data of the piezo forms and change it to another one. Why are these transformations so important? Because the manufacturers usually publish the materials data for D and S, while some FEM software’s require the piezoelectric matrix. The four possible forms of the piezoelectric constitutive equations are shown below: Deformation-Electric charge 𝑆𝑆 = 𝑠𝑠𝑒𝑒. 𝑇𝑇 + 𝑑𝑑𝑡𝑡. 𝐸𝐸 𝐷𝐷 = 𝑑𝑑. 𝑇𝑇 + 𝜖𝜖𝑇𝑇. 𝐸𝐸 Stress-Electric charge 𝑇𝑇 = 𝐶𝐶𝑒𝑒. 𝑆𝑆 − 𝑡𝑡𝑡𝑡. 𝐸𝐸 𝐷𝐷 = 𝑡𝑡. 𝑆𝑆 + 𝑡𝑡𝑆𝑆. 𝐸𝐸 Deformation-Stress 𝑆𝑆 = 𝑠𝑠𝐷𝐷. 𝑇𝑇 + 𝑟𝑟𝑡𝑡. 𝐷𝐷 𝐸𝐸 = −𝑟𝑟. 𝑇𝑇 + 𝑡𝑡𝑇𝑇−1. 𝐷𝐷 Stress-Potential gradient 𝑇𝑇 = 𝐶𝐶𝐷𝐷. 𝑆𝑆 − 𝑞𝑞𝑡𝑡. 𝐷𝐷 𝐸𝐸 = −𝑞𝑞. 𝑆𝑆 + 𝑡𝑡𝑆𝑆−1. 𝐷𝐷

1.5.4 Analysis of the Constitutive Relations

Previously we have shown examples of constitutive equation, but the properties of the materials have not been analyzed. This characteristic will be discussed in the following section and the piezoelectric materials will be taken into consideration. The piezoelectric matrix with orthotropic behavior and polarization along 𝑡𝑡₃ direction is shown

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⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢_𝜎𝜎𝜎𝜎1₂ 𝜎𝜎3 𝜏𝜏12 𝜏𝜏13 𝜏𝜏23 𝐷𝐷1 𝐷𝐷2 𝐷𝐷3⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ = ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢𝐶𝐶11𝐸𝐸 𝐶𝐶12𝐸𝐸 𝐶𝐶13𝐸𝐸 𝐶𝐶12𝐸𝐸 𝐶𝐶22𝐸𝐸 𝐶𝐶23𝐸𝐸 𝐶𝐶13𝐸𝐸 0 0 0 0 0 𝑡𝑡31 𝐶𝐶23𝐸𝐸 0 0 0 0 0 𝑡𝑡32 𝐶𝐶33𝐸𝐸 0 0 0 0 0 𝑡𝑡33 0 0 0 0 0 0 0 𝐶𝐶44𝐸𝐸 0 0 0 𝑡𝑡24 0 0 0 𝐶𝐶55𝐸𝐸 0 𝑡𝑡15 0 0 0 0 0 𝐶𝐶66𝐸𝐸 0 0 0 0 0 −𝑡𝑡31 0 0 −𝑡𝑡32 0 0 −𝑡𝑡15 0 𝑡𝑡11𝜖𝜖 0 0 0 −𝑡𝑡24 0 0 0 𝑡𝑡22𝜖𝜖 0 −𝑡𝑡33 0 0 0 0 0 𝑡𝑡33𝜖𝜖 _⎦⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢𝜖𝜖_𝜖𝜖1₂ 𝜖𝜖3 𝛾𝛾12 𝛾𝛾13 𝛾𝛾23 𝐸𝐸1 𝐸𝐸2 𝐸𝐸3⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ (1.14)

The transversely isotropic properties are characterized by the piezoelectric coupling: 𝑡𝑡32= 𝑡𝑡32 𝑡𝑡15 = 𝑡𝑡24 (1.15)

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Chapter 2

Description of the Specimens

This section will be described about the specimens (bending) selected for the Numerical simulation. In the first part a brief description of the MFC patches from the Smart Material Corp. is done. Then will be outlined the main geometric properties and features of the various samples.

2.1 MFC Piezo-patches from Smart Material Corp.

All the information below can be obtained in the Smart Material Corp. website. The Macro Fiber Composite (MFC) was invented by NASA in 1966. Smart Material started commercializing the MFC as the licensed manufacturer and distributor of the patented invention worldwide in 2002. The MFC consists of rectangular piezo-ceramic rods sandwiched between layers of adhesive, electrodes and polyimide film. The electrodes are attached to the film in a interdigitated pattern which transfers the applied voltage directly to and from the ribbon shaped rods. As a thin surface it can be applied (normally bonded) to various types of structure or embedded in a composite structure. If a voltage is applied, it will bend or distort the material, counteract vibrations or generate them. If no voltage is applied, it can work as a very sensitive strain gauge, sensing deformations, noise and vibrations. The MFC is also an excellent device to harvest energy from vibrations. In our applications P1 type MFC is used for the bending specimens. The P1 exploit the 𝑑𝑑₃₃ effect for actuation and elongate up to 1800ppm if they operate at maximum voltage: -500V, 1500V. The schematics of the MFC are shown in Figure11.

For our purpose we have selected the model M8528-P1 patch for the bending specimen. 2.2 Bending Specimen: Description

It is made up of 6 piezoelectric patches, three of these bonded patches are on the upper surface of the composite specimen and the other three patches are glued on the lower one. The piezoelectric material selected for this work is Macro Fiber Composite MFC (Smart Material) Figure 11.

Table 1 MFC Patch Main Features

M8528-P1 Active length(mm) 85 Active width(mm) 28 Overall length(mm) 103 Overall width(mm) 35 Capacitance(nF) 6.58 Free strain(ppm) 1800 Blocking Force(N) 454

Maximum operating positive Voltage(V) +1500 Minimum operating negative Voltage(V) -500

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Figure 11 Structural Design of the MFC Actuators.

The MFC has been chosen as an alternative actuator material because it has a much better profile if compared with other piezoelectric materials. In fact, MFC is using a diced piezo-wafer inside instead of the mono-layer of round ceramic fibers. Due to this, the coupling of the electric field into the ceramic material is the best and based on the rectangular shape of the fibers. The fill factor in the active cross-section is much more closely mesh compared with others. This leads to a better actuator strain and, due to a spherical electrode pattern, the MFC can be modified to make an anisotropic generation of a torsional structure deformation. For the above-mentioned layout, the fiber orientation of the MFC patches change. The bending specimen fiber orientation will be equal to 0°, compared with same direction, Figure 12

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Figure 12 MFC Fiber Orientation.

In the above models we have considered M8528-P1 which is bending specimen, the 𝑑𝑑₃₃ value, corresponding to the mechanical-electrical coupling effect, has been inserted in the field corresponding to the 𝑑𝑑₃₁ constant. The value is been considered because of the application of the voltage boundary conditions: the electric potential gradient is applied along the thickness of the patch (direction 3), the strain however remains in-plane (direction 1), Figure 13. For this material, the maximum operational positive voltage is equal to 1500 V, the maximum negative voltage is equal to -500 V.

Figure 13 Voltage Loading Conditions to the Active Volumes.

Two different composite materials have been considered for the laminate for the bending specimen.

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1. Graphite fiber / epoxy resin laminate; 2. Glass Fiber / epoxy resin laminate.

Each laminate consists of four fabric plies with two different fiber orientations: 0°/90° and 45°_/₋₄₅°_{respect to the length of the specimen. Table 2.2 shows the fiber orientation for the}

different plies.

Ply N. Sequence N.1 Sequence N.2 Sequence N.3 Sequence N.4

1 +45/-45 0/90 0/90 +45/-45

2 0/90 +45/-45 0/90 +45/-45

3 0/90 +45/-45 0/90 +45/-45

4 +45/-45 0/90 0/90 +45/-45

Table 2 Fiber Orientation for Different Plies

2.2.1 Bending Specimen

Bending deformation is been obtained along the length of the laminate: this can be done using MFC patches with elongation and applied contraction effects, respectively, on upper and lower surfaces of the laminate itself. The piezoelectric fiber direction must be set along the length of the substrate laminate: for this reason, MFC M85258-P1 , at 0° orientation, have been used. The experimental tests which were done in the previous years have been provided information on the deformation of the specimen along its length and max displacement will occur at the tip section of the specimen when an electric potential is applied on the MFC patches. A vertical load, on the tip of the specimen, must be applied in order to obtain information on the static or quasi-static response and the forces applied on the specimen. In the following Figure 14 and Figure 15, the geometry of the bending’s sample is shown.

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Figure 15 Geometry Dimensions of the Bending Specimen.

Below in Figure 16 we can see the cross-sectional view of the specimen

Figure 16 Cross-sectional view of Bending Specimen.

2.2.1.1 Bending Specimen – Honeycomb Core

Specimen with Honeycomb as core is not been manufactured, development phase is still running for the production. Here the Bending specimen which we are having is been modified by inserting a gap between four substrates, splitting them up by a thickness which is related to honeycomb and then the gap is filled with orthotropic material aluminum honeycomb.

Honeycomb geometry is similar to the bending specimen, but we have ignored the lower MFC patches, kapton and resin which are placed on the bottom face of lower substrate.

Figure 17 and Figure 18 describes the Bending specimen with core as honeycomb, includes 2 different layup processes with similar geometry to the bending specimen.

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Figure 17 Cross Sectional view of Four-Layer Specimen with Honeycomb Core.

Figure 18 Cross Sectional view of Three-layer Specimen with Honeycomb Core.

The dimensions of the components for both bending specimen and bending specimen with honeycomb core are same and they are as follows:

1. Substrate laminate (Face Sheets) = 400 mm × 45 mm × 1 mm 2. MFC = 85 mm × 28 mm

3. Honeycomb = 400 mm × 45 mm × 5 mm

2.3 Material Properties

The properties of the different material used in this work have been reported below:

Graphite/epoxy

Ε1 = Ε2 = Ε3 = 67.07 𝐺𝐺𝑃𝑃𝑑𝑑

𝜈𝜈12= 𝜈𝜈23 = 𝜈𝜈13= 0.042

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Glass/epoxy Ε1 = Ε2 = Ε3 = 25.27 𝐺𝐺𝑃𝑃𝑑𝑑 𝜈𝜈12= 𝜈𝜈23 = 𝜈𝜈13= 0.119 𝐺𝐺12= 𝐺𝐺13= 𝐺𝐺23= 4.83 𝐺𝐺𝑃𝑃𝑑𝑑 MFC Ε11 = 30.34 𝐺𝐺𝑃𝑃𝑑𝑑 Ε22= Ε33= 15.86 𝐺𝐺𝑃𝑃𝑑𝑑 𝜈𝜈12= 0.31 𝜈𝜈23= 𝜈𝜈13 = 0.16 𝐺𝐺12= 𝐺𝐺13= 𝐺𝐺23= 5.51 𝐺𝐺𝑃𝑃𝑑𝑑 𝐷𝐷11 = 𝐷𝐷22 = 𝐷𝐷33= 1.63802𝑡𝑡 − 8 _𝑚𝑚𝐹𝐹 𝑀𝑀𝑑𝑑𝑡𝑡𝑡𝑡𝑓𝑓𝑖𝑖𝑑𝑑𝑑𝑑 𝐷𝐷𝑖𝑖𝑡𝑡𝑑𝑑𝑡𝑡𝑑𝑑𝑡𝑡𝑓𝑓𝑖𝑖𝑑𝑑 𝑃𝑃𝑓𝑓𝑓𝑓𝑑𝑑𝑡𝑡𝑓𝑓𝑡𝑡𝑖𝑖𝑡𝑡𝑠𝑠 𝑑𝑑31= −4.6𝑡𝑡 − 10𝑚𝑚_𝑣𝑣 𝑀𝑀𝑑𝑑𝑡𝑡𝑡𝑡𝑓𝑓𝑖𝑖𝑑𝑑𝑑𝑑 𝑑𝑑𝑖𝑖𝑡𝑡𝑝𝑝𝑓𝑓𝑡𝑡𝑑𝑑𝑑𝑑𝑠𝑠𝑡𝑡𝑖𝑖𝑑𝑑 𝑠𝑠𝑡𝑡𝑓𝑓𝑑𝑑𝑖𝑖𝑖𝑖 𝑑𝑑𝑓𝑓 − 𝑡𝑡𝑓𝑓𝑓𝑓𝑡𝑡𝑑𝑑𝑖𝑖𝑡𝑡𝑖𝑖𝑡𝑡

Honeycomb – obtained from 1.4 & 1.5 Density = 20 𝑘𝑘𝑟𝑟/𝑑𝑑3 Ε11 = Ε22= 1 𝑀𝑀𝑃𝑃𝑑𝑑 Ε33= 100 𝑀𝑀𝑃𝑃𝑑𝑑 𝜈𝜈12= 0.3 𝜈𝜈23= 0.035 𝜈𝜈13= 0.0035 𝐺𝐺12= 1 𝑀𝑀𝑃𝑃𝑑𝑑 𝐺𝐺23= 329 𝑀𝑀𝑃𝑃𝑑𝑑 𝐺𝐺13= 128 𝑀𝑀𝑃𝑃𝑑𝑑

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Chapter 3

Finite Element Analysis Software – ANSYS

ANSYS is a general-purpose software, used to simulate interactions of all disciplines of physics, structural, vibration, fluid dynamics, heat transfer and electromagnetic for engineers. It enables to simulate tests in virtual environment working conditions before manufacturing prototypes of products. Furthermore, determining and improving weak points, computing life and foreseeing probable problems are possible by 3D simulations in virtual environment. ANSYS can work integrated with other used engineering software on desktop by adding CAD and FEA connection modules.

3.1 Sketching and Modelling

ANSYS Workbench combines access to ANSYS applications with utilities that manage the product workflow. Applications that can be accessed from Workbench include: ANSYS DesignModeler (for geometry creation); ANSYS Meshing (for mesh generation); ANSYS Polyflow (for setting up and solving computational fluid dynamics (CFD) simulations, where viscous and viscoelastic flows play an important role); and ANSYS CFD-Post (for post processing the results).

In Workbench, a project is composed of a group of systems. The project is driven by a schematic workflow that manages the connections between the systems. From the schematic, you can interact with workspaces that are native to Workbench, such as Design Exploration (parameters and design points), and you can launch applications that are data-integrated with Workbench. It allows you to construct projects composed of multiple dependent systems that can be updated sequentially based on a workflow defined by the project schematic.

Ansys DesignModeler is selected to create geometry. Through the available data discussed in the chapter two the following sketch and models are created.

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A finite element model of the structure was created. Initially, from the available data discussed in chapter two regarding the bending specimen dimensions where been analyzed. A rough outline lead to sophisticated sketch for the bending specimen. This obtained sketch gave path to wireframe model which is been considered as base wireframe model for both bending specimen and bending specimen with honeycomb. To reduce the complexity for modelling honeycomb structure, it is considered as orthotropic beam. Figure 20 describes the wireframe model for the specimen which will be considered for the complete rest of the work.

Figure 21 Wireframe Model for Bending Specimen.

Individual sketches where made in order to merge them together for obtaining a complete single model. Intentionally to obtain single model, similar sketches by parts where done. The obtained wireframe is extruded to get solid part.

The dimensions of the samples are as follows: -

Sample 1 (Bending specimen without honeycomb)

• Ply (Substrate) = 400 mm × 45 mm × 1mm (thickness 4 ply × 0.25mm = 1mm) • Bonding resin = 305 mm × 35 mm × 0.075mm (thickness = 0.075mm)

• MFC (Active area) = 85 mm × 28 mm ( thickness = 0.3 mm)

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Sample 2 (Bending specimen with honeycomb 1)

• Face sheets (4) = 400 mm × 45 mm × 1 mm (thickness 4 Face Sheets × 0.25mm =1mm) • Bonding resin = 305 mm × 35 mm × 0.075 mm (thickness = 0.075 mm)

• MFC (Active area) = 85 mm × 28 mm (thickness = 0.3mm)

• Kapton (MFC passive area) = 305 mm × 35mm × 0.3mm (thickness = 0.3mm) • Honeycomb = 400 mm × 45 mm × 5mm (thickness = 5mm)

Sample 3 (Bending Specimen with honeycomb 2)

• Face sheets(4)= 400 mm × 45 mm × 0.5 mm (thickness 4 Face Sheets × 0.125mm = 0.5 mm)

• Bonding resin = 305 mm × 35 mm × 0.075 mm (thickness = 0.075 mm) • MFC (Active area) = 85 mm × 28 mm (thickness = 0.3mm)

• Kapton (MFC passive area) = 305 mm × 35mm × 0.3mm (thickness = 0.3mm) • Honeycomb = 400 mm × 45 mm × 3 mm (thickness = 3mm)

Sample 4 (Bending Specimen with honeycomb 2 – Both Sides MFC)

• Face sheets(3)= 400 mm × 45 mm × 0.375 mm (thickness 3 Face Sheets × 0.125mm = 0.375 mm)

• Bonding resin = 305 mm × 35 mm × 0.075 mm (thickness = 0.075 mm) • MFC (Active area) = 85 mm × 28 mm (thickness = 0.3mm)

• Kapton (MFC passive area) = 305 mm × 35mm × 0.3mm (thickness = 0.3mm) • Honeycomb = 400 mm × 45 mm × 3 mm (thickness = 3mm)

To achieve the conformal mesh necessary in a way that each node matches exactly with the node of other elements, the model has been divided into a lot of rectangular geometrical bodies. All these geometric bodies, subsequently, have been tied up in a unique pat, so, when the software transfers the data of the geometry to the mechanical environment, through the share topology procedure, the nodes in contact merge together. Following figures describes the created samples with number of parts and bodies

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Figure 22 Sample 1 Model.

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Figure 24 Sample 3 Model.

Figure 25 Sample 4 Model

In this environment the coordinate systems, to define the fiber orientation of each single ply, honeycomb and to define the fiber orientation of the MFC element have been created. The

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definition of the right coordinate system for the MFC and the fiber composite laminate, honeycomb is necessary to define the behavior of the materials under the voltage loads.

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Chapter 4

FEM Analysis

A great part of the work done aims to validate the FEM model. In mathematics, the finite element method (FEM) is a numerical technique used to find approximate solutions to the boundary value problems for the partial differential equations. It uses subdivisions of a whole problem domain into simpler parts, called finite elements, and variational methods from the calculus of variations to solve the problem by minimizing an associated error function. FEM encompasses several methods to connect a lot simple element equation over many small subdomains, named finite elements, to approximate a more complex equation over a larger domain. It has been used the finite element method following this standard procedure:

• Pre-processing phase: the data needed to perform the analysis are input by the user. Activities driven by the user in the preprocessing phase include the selection of coordinate system, the element types, the definition of real constants, the material properties, the coupling and constraint equation, as well as the creation and meshing of the solid models and the manipulation of nodes and elements.

• Solution phase: the computer takes over and solve the simultaneous equations that the finite element method generates. The element solution is usually calculated at the elements integration points. The program writes the results in the database as well as the results file;

• Post processing phase: the display, the results of the analysis, the displacements, the stresses, the strain etc.. have been set up in the preprocessing phase and computed in the solution phase. The solution results can be output as a graphic display and/ or table report. The amount and the type of data available are controlled by the type of analysis that has been performed. The options have been defined by the user in the solution phase.

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Figure 27 Tree Outline of Project

4.1 Materials and Geometry

In this section the piezoelectric behavior has been examined through the use of the ANSYS software. As discussed in the chapter three, models of several samples were created, and this section describes the material properties need to be given to those samples containing substrates, resin, kapton, mfc and honeycomb.

The material properties which are briefly discussed in chapter two are assigned for the sample 1,2,3,4. Material properties are obtained from various studies, namely to mention from the research papers of Prof. Chiarelli and composite materials of R Jones.

The following tabular form specifies the material properties assigned to all the samples.

1 Property Value-Unit

2 Density 2038.6 𝐾𝐾𝑟𝑟/𝑑𝑑3

3 Orthotropic Elasticity

4 Young’s Modulus in X direction 2.527e+10 Pa 5 Young’s Modulus in Y direction 2.527e+10 Pa 6 Young’s Modulus in Z direction 2.527e+10 Pa

7 Poisson’s Ratio XY 0.119

8 Poisson’s Ratio YZ 0.119

9 Poisson’s Ratio XZ 0.119

10 Shear Modulus XY 4,83e+09 Pa

11 Shear Modulus YZ 4,83e+09 Pa

12 Shear Modulus XZ 4,83e+09 Pa

Table 3 Glass/epoxy Material Properties Assigned for Samples 1,2,3,4

2 Density 1852 𝐾𝐾𝑟𝑟/𝑑𝑑3

10 Shear Modulus XY 4.78e+09 Pa

11 Shear Modulus YZ 4.78e+09 Pa

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10 Shear Modulus XY 5.51e+09 Pa

11 Shear Modulus YZ 5.51e+09 Pa

12 Shear Modulus XZ 5.51e+09 Pa

Table 5 MFC Material Properties Assigned for Samples 1,2,3,4

4 Young’s Modulus in X direction 1e+06 Pa

5 Young’s Modulus in Y direction 1e+06 Pa

6 Young’s Modulus in Z direction 100e+06 Pa

10 Shear Modulus XY 1e+06 Pa

11 Shear Modulus YZ 329e+06 Pa

12 Shear Modulus XZ 128e+06 Pa

Table 6 Honeycomb Material Properties Assigned for Samples 1,2,3,4

2 Density 1320 𝑘𝑘𝑟𝑟/𝑑𝑑3

3 Isotropic Elasticity

4 Young’s Modulus 2.5e+09 Pa

5 Poisson’s Ratio 0.34

6 Bulk Modulus 2.6042e+09 Pa

7 Shear Modulus 9.3284e+08 Pa

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2 Density 833.4 𝑘𝑘𝑟𝑟/𝑑𝑑3

3 Isotropic Elasticity

4 Young’s Modulus 1e+10 Pa

5 Poisson’s Ratio 0.4

6 Bulk Modulus 1.6667e+10 Pa

7 Shear Modulus 3.5714e+09 Pa

Table 8 Resin Material Properties Assigned for Samples 1,2,3,4

4.2 Mechanical Models

In our model it is most important to specify right element type. We have considered two types of elements, namely,

1. Solid226 2. Solid186

4.2.1 Solid226

The element has twenty nodes with up to five degrees of freedom per node. Structural capabilities include elasticity, plasticity, hyperelasticity, viscoplasticity, large deflection and prestress effects. Solid226 has following capabilities:

• Structural-Thermal • Piezoresistive • Electroelastic • Piezoelctric • Thermal-Electric • Thermal-piezoelctric

As Solid226 holds piezoelectirc capabilities, we consider the element. Assigning the element type to the concerned body MFC requires using a proper command. In ANSYS the piezoelectric model requires permittivity (or dielectric constants); the piezoelectric matrix and the elastic coefficient matrix have to be specified as material properties. The orthotropic dielectric matrix at constant strain ⌊𝜀𝜀𝑠𝑠⌋ uses the electrical permittivity’s (input as PERX, PERY, and PERZ on the MP commands) and it is of this form:

⌊𝜀𝜀𝑠𝑠⌋ = �

𝜀𝜀11 0 0

0 𝜀𝜀22 0

0 0 𝜀𝜀33

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In this case 𝜀𝜀₁₁= 𝜀𝜀₂₂ = 𝜖𝜖₃₃ and the values have been calculated as the ratio between the material electrical permittivity (𝜀𝜀_𝑚𝑚= 1.63802𝑡𝑡 − 8 𝐹𝐹

𝑚𝑚) and the vacuum electrical

permittivity (𝜀𝜀₀ = 8.85418781762𝑡𝑡 − 12 𝐹𝐹

𝑚𝑚).

PERX = PERY = PERZ = 𝜀𝜀𝑚𝑚

𝜀𝜀0 = 1850 (4.2)

ANSYS automatically converts the piezoelectric strain matrix ⌊𝑑𝑑⌋ to a piezoelectric stress matrix ⌊𝑡𝑡⌋ using the elasticity matrix ⌊𝑑𝑑⌋, with the following formula:

⌊𝑡𝑡⌋ = ⌊𝑑𝑑⌋. ⌊𝑑𝑑⌋ (4.3) Where: ⌊𝑡𝑡⌋ = ⎣ ⎢ ⎢ ⎢ ⎢ ⎢𝑡𝑡_𝑡𝑡11₂₁ 𝑡𝑡_𝑡𝑡12₂₂ 𝑡𝑡_𝑡𝑡13₂₃ 𝑡𝑡31 𝑡𝑡41 𝑡𝑡51 𝑡𝑡61 𝑡𝑡32 𝑡𝑡42 𝑡𝑡52 𝑡𝑡62 𝑡𝑡33 𝑡𝑡43 𝑡𝑡53 𝑡𝑡63⎦ ⎥ ⎥ ⎥ ⎥ ⎥ (4.4) ⌊𝑑𝑑⌋ = ⎣ ⎢ ⎢ ⎢ ⎢ ⎢𝑑𝑑_𝑑𝑑11 𝑑𝑑12 𝑑𝑑13 21 𝑑𝑑22 𝑑𝑑23 𝑑𝑑31 𝑑𝑑41 𝑑𝑑51 𝑑𝑑61 𝑑𝑑32 𝑑𝑑42 𝑑𝑑52 𝑑𝑑62 𝑑𝑑33 𝑑𝑑43 𝑑𝑑53 𝑑𝑑63⎦ ⎥ ⎥ ⎥ ⎥ ⎥ (4.5)

Modelling of Sandwich Composite Specimen with MFC actuators: Static Simulation on Bending.