Macro-Mechanics - A271

 

Macro-Mechanics
Foundational knowledge article
Document Type	Article
Document Identifier	271
Tags	Mechanical properties
Prerequisites	Composite properties Micro-Mechanics

Introduction

Macro-mechanics or laminate theory is used when two or more laminae are bonded together (a laminate) to determine the resultant properties of such a composite. By using the inherent properties of each lamina, see Micro-Mechanics, classical laminate theory is used to determine individual ply stresses and strains as well as the global strains and global engineering constants for a load applied at any angle. Laminate theory is a macro-mechanical analysis of the orthotropic fibrous plies stacked together in different orientations to get the desired characteristics of a composite laminate.

Level I

Level II

Beam Theory

Stress distribution in a laminate as a result of linear strain distribution and non-linear E modulus.

To construct a composite part many plies of lamina are stacked together to form a laminate. With the background information above on how a single lamina performs under various stresses, this can be combined to predict the behavior of a composite laminate. Each ply is assumed to be a homogenous and orthotropic sheet such that laminate theory can be applied to the construction as a whole. Classical laminate theory is used to predict thin beams and plates as it neglects shear deflection in the laminate, meaning the thickness is small compared to the lateral dimensions. This is based on the Euler-Bernoulli beam theory or classic beam theory. It is also assumed that there is perfect bonding between the laminae, such that they cannot slide over one another. Classic beam theory dictates that the total deflection in a beam is due to both the bending and shear deflection within the beam. The resultant strain will be a linear distribution through the cross section of the beam. In an isotropic beam this will yield a linear stress distribution as well, however for a laminate the E modulus is not constant throughout and will result in a staggered stress distribution.

Laminate Codes

Select examples of laminate codes and corresponding illustrations.

Much like the convention of the laminate coordinate system, there is also a convention of describing a laminate in terms of each ply and their direction. Each layer is identified by its location and angle relative to a reference point and direction. \([0/-45/90/60/30]_{T}\) Each number represents a ply and its angle, plies are separated by ‘/’ and the first ply is the top ply. There can be a subscript at the end describing the type and amount of repetition: ‘T’ means this is the total laminate, ‘S’ means the laminate is symmetrical and there may be a number indicating repetitions. There may be subscripts at individual plies meaning repetition of the ply. Lastly an over bar at the last ply means it is not repeated in the symmetry, but is the centre ply of the laminate. See figure for some examples.

Coupling of Loads

A: Axial in longitudinal and transverse directions. B: Bending and axial (eliminated by symmetric laminate). C: Twisting and axial (eliminated by symmetric laminate). D: Twisting and shear (eliminated by symmetric laminate). E: Axial and shear (eliminated by balanced laminate). F: Bending in longitudinal and transverse directions.

Moving from a beam to a plate laminate it introduces new coupling effects seen by the different orthotropic layers due to moment and forces in the y direction. Each lamina will behave differently depending on the fiber direction when stress is applied, a common way to mitigate adverse coupling loads is to create a symmetrical and balanced laminate. Below are examples of each load coupling.

Beam Theory

To construct a composite part many plies of lamina are stacked together to form a laminate. With the background information above on how a single lamina performs under various stresses, this can be combined to predict the behavior of a composite laminate. Each ply is assumed to be a homogenous and orthotropic sheet such that laminate theory can be applied to the construction as a whole. Classical laminate theory is used to predict thin beams and plates as it neglects shear deflection in the laminate, meaning the thickness is small compared to the lateral dimensions. This is based on the Euler-Bernoulli beam theory or classic beam theory. It is also assumed that there is perfect bonding between the laminae, such that they cannot slide over one another. Lastly there is the Kirchhoff assumption, which states that the in-plane displacements are linear functions of the thickness. This means that the interlaminar shear strains are negligible: \(\varepsilon_{xz} = \varepsilon_{yz} = 0\)

Derivation of Strain Equation

This allows us to estimate laminate behaviour with a 2D analysis of the midplane, giving the following strain-displacement relationships, where u, v and w are displacements in the x, y and z directions respectively: \[\begin{aligned} &\varepsilon_{x}=\frac{\partial u}{\partial x} \quad \varepsilon_{x y}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\\ &\varepsilon_{y}=\frac{\partial v}{\partial y} \quad \varepsilon_{x z}=\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}\\ &\varepsilon_{z}=\frac{\partial w}{\partial z} \quad \varepsilon_{y z}=\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y} \end{aligned}\]

Due to the Kirchhoff assumption of in plane displacement being a linear function of the thickness, then: \(u=u_{0}(x, y)+z F_{1}(x, y) \quad v=v_{0}(x, y)+z F_{2}(x, y)\) Where \(F_{1}\) and \(F_{2}\) are determined by the assumption that interlaminar shears are equal to zero: \[\begin{aligned} \varepsilon_{x z} &=F_{1}(x, y)+\frac{\partial w}{\partial x}=0 \\ \varepsilon_{y z} &=F_{2}(x, y)+\frac{\partial w}{\partial y}=0 \end{aligned}\] Such that: \[F_{1}(x, y)=-\frac{\partial w}{\partial x} \quad \text { and } \quad F_{2}(x, y)=-\frac{\partial w}{\partial y}\] Substituting the newly defined \(F_{1}(x, y)\) and \(F_{2}(x, y)\) back into the linear functions and derive to obtain strain functions: \[\begin{aligned} &\varepsilon_{x}=\frac{\partial u}{\partial x}=\frac{\partial u_{0}}{\partial x}-z \frac{\partial^{2} w}{\partial x^{2}}=\varepsilon_{x}^{0}+z K_{x}\\ &\varepsilon_{y}=\frac{\partial v}{\partial y}=\frac{\partial v_{0}}{\partial y}-z \frac{\partial^{2} w}{\partial y^{2}}=\varepsilon_{y}^{0}+z K_{y}\\ &\varepsilon_{x y}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}=\frac{\partial u_{0}}{\partial y}+\frac{\partial v_{0}}{\partial x}-2 z \frac{\partial^{2} w}{\partial x \partial y}=\varepsilon_{x y}^{0}+z K_{x y} \end{aligned}\] The expression K captures the curvature of the beam and \([\varepsilon_{0}]\) is the midplane strain. We can now place the three expressions into matrix form, where the strain and K values are as such: \[\left[\begin{array}{c} \varepsilon_{x}^{0} \\ \varepsilon_{y}^{0} \\ \varepsilon_{xy}^{0} \end{array}\right]=\left[\begin{array}{c} \partial u_{0} / \partial x \\ \partial v_{0} / \partial y \\ \partial u_{0} / \partial y+\partial v_{0} / \partial x \end{array}\right]\] and \[\left[\begin{array}{c} K_{x} \\ K_{y} \\ K_{xy} \end{array}\right]=-\left[\begin{array}{c} \partial^{2} w / \partial x^{2} \\ \partial^{2} w / \partial y^{2} \\ 2 \partial^{2} w / \partial x \partial y \end{array}\right]\] And the final strain equation comes out to be: \[\left[\begin{array}{l} \varepsilon_{x} \\ \varepsilon_{y} \\ \varepsilon_{xy} \end{array}\right]=\left[\begin{array}{l} \varepsilon_{x}^{0} \\ \varepsilon_{y}^{0} \\ \varepsilon_{xy}^{0} \end{array}\right]+z\left[\begin{array}{l} K_{x} \\ K_{y} \\ K_{xy} \end{array}\right]\]

Strain and Stress Distribution Through Thickness of a Beam

Stress distribution in a laminate as a result of linear strain distribution and non-linear E modulus.

As established above by the Kirchhoff assumption, the strain through a laminate beam is linear, however the stress is not. This is due to that each lamina in the ply stack has different properties due to its orthotropic nature and fiber angles to the loading direction giving different Young’s modulus values, see figure to the right. The stress in a single ply, k, in a composite can be expressed from Micro-Mechanics as: \[ [\sigma]_{k}=[\bar{Q}]_{k}[\varepsilon]_{k}\] Where, as seen above, the strain is expressed as \[ [\varepsilon]=\left[\varepsilon^{0}\right]+z[K]\] Giving a stress distribution expression of: \[ [\sigma]_{k}=[\bar{Q}]_{k}\left[\varepsilon^{0}\right]+z[\bar{Q}]_{k}[K]\]

Stiffness Matrices and Stress Analysis

When combining all the plies in a beam or plate there will be a resultant force and a resultant moment in the laminate. We are considering a laminate in a state of plane stress and thus the relevant stresses, as previously considered, will be \(\sigma_{x}, \sigma_{y}\) and \(\sigma_{xy}\). Labelling the resultant forces as N and moments as M, they can be defined at the midplane of the laminate by the following integrations: \[\begin{aligned} &N_{x}=\int_{-h / 2}^{k / 2} \sigma_{x} d z \\ &N_{y}=\int_{-h / 2}^{h / 2} \sigma_{y} d z \quad\\ &N_{xy}=\int_{-h / 2}^{h / 2} \sigma_{xy} d z \end{aligned} \quad \quad \begin{aligned} M_{x} &=\int_{-h / 2}^{h / 2} \sigma_{x} z d z \\ M_{y} &=\int_{-h / 2}^{h / 2} \sigma_{y} z d z \\ M_{x y} &=\int_{-h / 2}^{h / 2} \sigma_{xy} z d z \end{aligned}\] The resultant forces have the units of force per unit length and are positive in the direction of the stress components. The moments are defined as the moments produced by the stresses with respect to the midplane. Using the x direction as an example for the three directions, substituting in strain and the Young’s modulus gives an expression: \[N=\int_{-t / 2}^{t / 2} \sigma_{x} d z=\int_{-t / 2}^{t / 2} E \varepsilon_{x} d z=\int_{-t / 2}^{t / 2} E\left(\varepsilon_{0}+z \kappa\right) dz\] Simplifying the integral terms into expressions of A and B gives: \[N=\left(\int_{-t / 2}^{t / 2} E d z\right) \varepsilon_{0}+\left(\int_{-t / 2}^{t / 2} E z d z\right) \kappa \rightarrow N=A\varepsilon_{0}+B\kappa\] Doing the same for the moment also gives a B term and a similar D term: \[M=\left(\int_{-t / 2}^{t / 2} E z d z\right) \varepsilon_{0}+\left(\int_{-t / 2}^{t / 2} E z^{2} d z\right) \kappa \rightarrow M=B\varepsilon_{0}+D\kappa\] The forces and moments can now be combined into a simplified matrix, commonly referred to as the ABD matrix: \[\left\{\begin{array}{l} N \\ M \end{array}\right\}=\left[\begin{array}{ll} A & B \\ B & D \end{array}\right]\left\{\begin{array}{c} \varepsilon_{0} \\ \kappa \end{array}\right\}\] In this matrix each of the terms represent a type of stiffness property exhibited by the beam:

• \(A=\int_{\frac{t}{2}}^{\frac{t}{2}} E d z\) Extensional stiffness of the beam
• \(B=\int_{\frac{-t}{2}}^{\frac{t}{2}} E z d z\) Coupling stiffness of the beam
• \(D=\int_{\frac{-t}{2}}^{\frac{t}{2}} E z^{2} d z\) Bending stiffness of the beam.

Plate Theory

Figure depicting all the relevant resultant forces and moments acting on a thin plate.

When building upon the beam theory to include a full laminated plate, there are additional resultant forces and moments that need to be considered. This leads to each of the four terms in the ABD matrix becoming their own 3x3 matrix. Since the plate is thin, the through thickness forces and moments can be ignored (\(N_{z}\), \(N_{zx}\) & \(N_{yz}\), \(M_{x}\), \(M_{zx}\) & \(M_{yz}\) ). See figure for a depiction of all the relevant resultant forces and moments acting on a thin plate. The ABD matrix then becomes: \[\left\{\begin{array}{l} \{N\} \\ \{M\} \end{array}\right\}=\left[\begin{array}{ll} {[A]_{3 \times 3}} & {[B]_{3 \times 3}} \\ {[B]_{3 \times 3}} & {[D]_{3 \times 3}} \end{array}\right]\left\{\begin{array}{c} \left\{ \varepsilon^{0}\}\right. \\ \{\kappa\} \end{array}\right\}\] or: \[\left\{\begin{array}{l} N_{x} \\ N_{y} \\ N_{x y} \\ M_{x} \\ M_{y} \\ M_{x y} \end{array}\right\}=\left[\begin{array}{llllll} A_{11} & A_{12} & A_{16} & B_{11} & B_{12} & B_{16} \\ A_{21} & A_{22} & A_{26} & B_{21} & B_{22} & B_{26} \\ A_{16} & A_{26} & A_{66} & B_{16} & B_{26} & B_{66} \\ B_{11} & B_{12} & B_{16} & D_{11} & D_{12} & D_{16} \\ B_{21} & B_{22} & B_{26} & D_{21} & D_{22} & D_{26} \\ B_{16} & B_{26} & B_{66} & D_{16} & D_{26} & D_{66} \end{array}\right]\left\{\begin{array}{c} \varepsilon_{x}^{o} \\ \varepsilon_{y}^{o} \\ \gamma_{x y}^{o} \\ \kappa_{x} \\ \kappa_{y} \\ \kappa_{x y} \end{array}\right\}\]

Coupling of Loads and Types of Laminates

A: Axial in longitudinal and transverse directions. B: Bending and axial (eliminated by symmetric laminate). C: Twisting and axial (eliminated by symmetric laminate). D: Twisting and shear (eliminated by symmetric laminate). E: Axial and shear (eliminated by balanced laminate). F: Bending in longitudinal and transverse directions.

There different terms in the ABD matrix refer to different types of coupling effect experienced by the laminate. \(A_{ij}\) and \(D_{ij}\) for all \(i=j\) Express the different rigidities in the system: Axial, shear, bending and torsional. Refer to the figure for illustrations of the different coupling effect. Figure A refers to \(A_{11}\), which is the coupling of axial strains. Figure B refers to \(B_{11}\), \(B_{12}\) and \(B_{22}\), showing the coupling between bending and axial strains. Figure C refers to \(B_{16}\) and \(B_{26}\), which is the coupling of twisting and axial strains. Figure D refers to \(B_{66}\), which is the coupling of twisting and shear strains. All of the \(B\) terms are even functions of \(h_{k}\), this is due to that they are sums of terms from \(\bar Q_{ij}\) and the difference of the square of \(z\) terms for the top \(h_{k}\) and the bottom \(h_{k-1}\) for each ply. This means that the \(B\) terms are zero if there is an identical ply an equal distance below the midplane, thus these coupling effect can be eliminated by using a symmetric laminate.

Figure E refers to \(A_{16}\) and \(A_{26}\), which is the coupling of shear and axial strains. These terms rely on the angle of the laminate to give them a value and the terms are positive or negative depending on the angle. This means that to cancel out this effect there needs to be a ply in the laminate of equal and opposite angle to the ply, this is called a balanced laminate.

Lastly figure F refers to \(D_{12}\) and is the coupling of bending curvatures. Not pictured is the coupling of twisting and bending curvatures represented by \(D_{16}\) and \(D_{26}\). Much like the \(A\) terms, these terms are dependent on the angle of the plies, stemming from \(\bar Q_{16}\) and \(\bar Q_{26}\). Such that a balanced laminate will eliminate these coupling effects. However, the \(D_{16}\) and \(D_{26}\) terms are also dependent on the distance from the midplane, as it needs to be equal and opposite. This stems from the \(h^{3}_{k}\) and \(h^{3}_{k-1}\) from \(\bar Q\) that the \(D\) terms are relying on, specifically the difference between them. Meaning that in order for \(D_{16}\) and \(D_{26}\) to be zero, the \(B\) matrix will be non-zero. Below is an illustration summarizing the different effects discussed here.

Illustration of three different laminate builds that will eliminate coupling effects.

Strains in Laminates

If the forces are known, the strains and curvatures can be calculated directly by using and inverted ABD matrix. \[\left[\frac{\varepsilon^{0}}{K}\right]=\left[\begin{array}{ll} A^{\prime} & B^{\prime} \\ B^{\prime} & D^{\prime} \end{array}\right] \left[\frac{N}{M}\right]\] Where: \[\begin{aligned} {\left[A^{\prime}\right] } &=\left[A^{*}\right]+\left[B^{*}\right]\left[D^{*}\right]^{-1}\left[B^{*}\right]^{T} \\ {\left[B^{\prime}\right] } &=\left[B^{*}\right]\left[D^{*}\right]^{-1} \\ {\left[D^{\prime}\right] } &=\left[D^{*}\right]^{-1} \end{aligned}\]

Laminate Elastic Constants

Since a symmetric laminate has a B matrix of zero and does not experience coupling between extension and bending, the design process can be greatly simplified by determining elastic constants. This makes the design of the composite behave more like a traditional material and we can determine the constants by inverting the stress resultant midplane strain relation: \[\{\varepsilon^{0}\}=[A]^{-1}\{N\}=[a]\{N\}\] \([a]\)is referred to as the elastic compliance matrix, which we can use to determine the constants: \[\begin{aligned} &E_{x}=\frac{1}{2 h a_{11}} \quad \quad G_{x y}=\frac{1}{2 h a_{66}} \\ &E_{y}=\frac{1}{2 h a_{22}} \quad \quad v_{x y}=-\frac{a_{12}}{a_{11}} \end{aligned}\] Where \(h\) is the thickness of the laminate.

To summarize the mechanics process to follow, a flow chart has been developed illustrating the determination of engineering properties for a full laminate:

A flowchart illustrating the order of operation in order to determine the elastic properties of a laminate from the individual properties of lamina.

Stress Analysis in Laminates

Below is a systematic approach to calculating the stresses in laminates.

1. Find the stiffness matrix \([Q]\) for each material.

2. Find the transformed stiffness matrix \([\bar Q]\) for each layer.

3. Find the location of the top of the laminate \(z_{0}\) and the location of the bottom of each layer \(z_{k}\) with respect to the centre (mid-plane) of the laminate.

4. Using the \([\bar Q]\) matrices, find \([A]\), \([B]\), \([D]\).

5. A. If the loads are given, calculate the mid-plane strains and curvatures. \[\left[\frac{\varepsilon^{0}}{K}\right]=\left[\begin{array}{ll} A^{\prime} & B^{\prime} \\ B^{\prime} & D^{\prime} \end{array}\right] \left[\frac{N}{M}\right]\] 5. B. If the mid-plane strains and curvatures are known, find the applied forces and moments: \[\left\{\begin{array}{l} \{N\} \\ \{M\} \end{array}\right\}=\left[\begin{array}{ll} {[A]_{3 \times 3}} & {[B]_{3 \times 3}} \\ {[B]_{3 \times 3}} & {[D]_{3 \times 3}} \end{array}\right]\left\{\begin{array}{c} \left\{\varepsilon^{0}\}\right. \\ \{\kappa\} \end{array}\right\}\] 6. For locations of interest, say at location I in layer k, find the strains: \[\left\{\begin{array}{c} \varepsilon_{x} \\ \varepsilon_{y} \\ \gamma_{x y} \end{array}\right\}_{I}=\left\{\begin{array}{c} \varepsilon_{x}^{0} \\ \varepsilon_{y}^{0} \\ \gamma_{x y}^{0} \end{array}\right\}+Z_{I}\left\{\begin{array}{c} \kappa_{x} \\ \kappa_{y} \\ \kappa_{x y} \end{array}\right\}\] 7. Calculate the stresses at location I \[\left\{\begin{array}{l} \sigma_{x} \\ \sigma_{y} \\ \tau_{x y} \end{array}\right\}_{I}=[\bar{Q}]_{k}\left\{\begin{array}{l} \varepsilon_{x} \\ \varepsilon_{y} \\ \gamma_{x y} \end{array}\right\}_{I}\] 8. Calculate the lamina (local) stresses or strains: \[\left\{\begin{array}{l} \sigma_{1} \\ \sigma_{2} \\ \tau_{12} \end{array}\right\}_{I}=[T]_{k}\left\{\begin{array}{l} \sigma_{x} \\ \sigma_{y} \\ \tau_{x y} \end{array}\right\}_{I} \quad \quad\left\{\begin{array}{c} \varepsilon_{1} \\ \varepsilon_{2} \\ \gamma_{12} \end{array}\right\}_{I}=\left[T^{*}\right]_{k}\left\{\begin{array}{c} \varepsilon_{x} \\ \varepsilon_{y} \\ \gamma_{x y} \end{array}\right\}_{I}\]

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Engineered materials (designed to have specific properties) made from two or more constituent materials with different physical or chemical properties. The constituents remain separate and distinct on a macroscopic level within the finished structure.

A laminate whose in-plane shear coupling stiffness is zero. This occurs if all plies in a laminate that are not oriented parallel or perpendicular to the primary laminate axis (i.e. non 0 or 90 degree plies) occur only in ±θ pairs that have the same thickness and elastic properties.

The continuous material phase that binds the reinforcement together, maintains shape, transfers load, protects the reinforcement from environment and damage, and provides the composite support in compression.

Desirable characteristics:

• Moisture/chemical resistance
• Low density
• Processability