Three composite models in stress , strain and elastic modulus
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The document discusses three models for composite materials: 1) The parallel phase model where the dispersed phase is embedded in a continuous matrix phase. It determines the elastic modulus (E1) of the composite in the fiber direction based on the modulus and volume fraction of the fibers and matrix. 2) The series phase model where the rule of mixtures relates the composite modulus to the modulus and volume fraction of the matrix and dispersed materials. 3) The dispersed phase model, also called the Maxwell model, which is not described in detail.
Three composite models in stress , strain and elastic modulus
- 1. 1 Gaziantep University College of Engineering Civil Engineering Department Report on Three composite models in stress , strain and elastic modulus Submitted by Mohammed Layth Abbas Student No:201444956 Supervisor Assoc.Prof. Erhan Güneyisi CE 543 (composite Materials)
- 2. 2 Introduction A composite material is made by combining two or more materials – often ones that have very different properties. The two materials work together to give the composite unique properties. However, within the composite you can easily tell the different materials apart as they do not dissolve or blend into each other. Why use composites? The biggest advantage of modern composite materials is that they are light as well as strong. By choosing an appropriate combination of matrix and reinforcement material, a new material can be made that exactly meets the requirements of a particular application. Composites also provide design flexibility because many of them can be molded into complex shapes. The downside is often the cost. Although the resulting product is more efficient, the raw materials are often expensive. Some of the properties that can be improved by forming a composite material are:- Strength , stiffness , corrosion resistance , wear resistance , attractive ness , weight , fatigue life , temperature dependent behavior , thermal insulation . Unit cell models of the composites
- 3. 3 There are three types of the unit cell models composite material as indicated below 1- Parallel phase model Dispersed phaseMatrix phase For the parallel phase model Determination of E1 The first modulus to be determined is that one of the composite material in the 1- direction that is in the fiber direction from the figure below ….1ɛ1= Where ɛ1 applies for both the fibers and the matrix according to the basic assumption. Then if both constituent materials behave elasticity the stress in the fiber direction are .....m = Em F = EF1 The average stress 1 acts on cross sectional area A of the representative volume element , F acts on the cross sectional area of the fibers Af and m acts on the cross sectional area of of the matrix Am .The resultant force of the reprehensive volume element of the composite material is .…3P=1 A =f Af +=f Af
- 4. 4 By substitution of equation (2 ) in equation (3 )and recognition from micromechanics that ….41 = E11 Apparently …..5Em+E1= Ef but the volume friction of fiber and matrix can be written as Vf=Af/A Vm=Am/A ….6 Thus …..7EmVm+E1 = EfVf 2-Series phase model The modulus of the matrix material is Em and the modulus of the dispersed material is Ed where is the modulus of the composite material is E the volume friction of the constituents are Vm and Vd such that ….1Vm+Vd=1 Obviously any relationship for the composite modulus E must yield E=Em for Vm =1 and E = Ed for Vd=1 One of the simplest relationships that satisfies the foregoing restrictions is the rule of mixtures ….2E=EmVm+EdVd
- 5. 5 Where in the constituents of the composite material are presumed to contribute to the composite stiffness in direct proportion to their own stiffness and volume frictions. The rule of mixtures will be shown to provide an upper bound on the composite modulus E for the special case in which ….3Vm=Vd=V Another simple relationship between the constituent moduli results from the observation that the compliance of the composite material 1/E, must agree with the compliance of the matrix. 1/Em, when Vm=1 and with the compliance of the dispersed material when Vd= 1 the resulting rule of mixtures for compliance is ….4 3-Dispersed phase (Maxwell modell)