Finite Element Analysis Of Composite Layered Structures Connor Kaufmann – B. Sc. ‘14
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Transcript of Finite Element Analysis Of Composite Layered Structures Connor Kaufmann – B. Sc. ‘14
Finite Element AnalysisOf Composite Layered Structures
Connor Kaufmann – B. Sc. ‘14 Neola Putnam – M. Eng. ‘14 Ethan Seo – M. Eng. ‘14 Ju Hwan (Jay) Shin – B. Sc. ‘14
Cornell UniversitySibley School of Mechanical & Aerospace Engineering
Spring 2014 – Professor N. Zabaras
Objective Develop a linear 3D finite element analysis from scratch using MATLAB.
Consider a uniaxial loading of a symmetric laminate.
Verify the results with expected results, namely the state of out-of-plane stresses near the free edges.
Observe the effect of h-refinement (convergence of results through mesh refinements).
Perform a simple sanity check by doing a force reaction balance with the applied traction.
Compare the numerical results with a commercial FE software, or Ansys Composite PrepPost (ACP).
2
C. Kaufmann, N. Putnam, E. Seo, J. Shin
0 20 40 60 80 1000
2000
4000
6000
8000
10000
12000
14000
Stress_yz for [0 90]s laminate
Along y-direction [m] yz
[P
a]
𝜏 𝑦𝑧
𝑃0
3
Some stress contours!C. Kaufmann, N. Putnam, E. Seo, J. Shin
𝝈𝒙𝒙 𝝈𝒚𝒚 𝝈𝒛𝒛
𝝉𝒚𝒛 𝝉𝒙𝒛 𝝉𝒙𝒚
These results refer to a cross-ply laminate.
Overview of composite materials Composite materials are commonly used in aerospace structures to minimize mass.
We considered symmetric, unidirectional, fiber-reinforced composites.
Composite lamina (sheets) can be stacked to form high strength laminates.
Laminate stack-ups are characterized by the orientation angles of the fibers, and the materials used.
Anisotropy of the laminates allows one to tailor designs for stiffness and strength in specific directions.
4i.e.
C. Kaufmann, N. Putnam, E. Seo, J. Shin
Complexities of layered structures Composite materials can give a much more complicated mechanical response than monolithic materials.
Stress equilibrium must be satisfied in the laminate by way of interlaminar stresses.
Special care must be taken to consider the free-edge and free-corner effect in composite samples!
Non-intuitive effects, such as normal-shear coupling, can occur in anisotropic materials.
As a result, finite element analysis is often found useful for predicting the behavior of complicated composites.
5
Delamination
C. Kaufmann, N. Putnam, E. Seo, J. Shin
Normal-shear coupling
6
FEM Formulation Pre-processing:
Define the required size dimensions, material properties, and the laminate configuration. Discretize the model into finite elements. Consider a tri-linear hexahedron element. Apply any bias factor when discretizing (optional). Calculate the 3D elasticity matrix, . i.e. Compute the elemental matrices necessary in developing the stiffness equation. Specify the boundary conditions (includes the external load).
Processing: “Globalize” and assemble the local stiffness matrices and the local load vectors. Partition and rearrange the global stiffness equation. Solve for the nodal displacement field!
Post-processing: Compute the strain field by applying the kinematic equation (displacement ↔ strain). Compute the stress field by applying the constitutive equation (Hooke’s Law).
C. Kaufmann, N. Putnam, E. Seo, J. Shin
7
Pre-processing (1/7) Define the required size dimensions, material properties, and the laminate configuration. Discretize the model into finite elements. Consider a 8-noded hexahedron, or tri-linear element (3 translational DOF per node). Apply any bias factor when discretizing (optional). Calculate the 3D elasticity matrix, . i.e. Compute the elemental matrices necessary in developing the stiffness equation. Specify the boundary conditions (includes the external load).
Define the coordinate axes, based on fiber orientations. x-axis: longitudinal direction y-axis: transverse direction z-axis: normal direction (or thickness direction)
Specify the size dimensions. , , and
Specify the material properties. , , , , , , , , and
Specify the fiber orientations of the off-axis plies.
C. Kaufmann, N. Putnam, E. Seo, J. Shin
𝑵 𝒙
8
Pre-processing (2/7) Define the required size dimensions, material properties, and the laminate configuration. Discretize the model into finite elements. Consider a 8-noded hexahedron, or tri-linear element (3 translational DOF per node). Apply any bias factor when discretizing (optional). Calculate the 3D elasticity matrix, . i.e. Compute the elemental matrices necessary in developing the stiffness equation. Specify the boundary conditions (includes the external load).
Store the global nodes as local nodes for each element.
Adhere to the given node-numbering scheme to ensure that the determinant of the Jacobian matrix is positive. Guarantee invertible mapping to natural coordinate system.
C. Kaufmann, N. Putnam, E. Seo, J. Shin
9
Pre-processing (3/7) Define the required size dimensions, material properties, and the laminate configuration. Discretize the model into finite elements. Consider a tri-linear hexahedron element. Apply any bias factor when discretizing (optional). Calculate the 3D elasticity matrix, . i.e. Compute the elemental matrices necessary in developing the stiffness equation. Specify the boundary conditions (includes the external load).
Use an eight-noded hexahedron element.
Each node has three translational degrees-of-freedom. , , and
C. Kaufmann, N. Putnam, E. Seo, J. Shin
10
Pre-processing (4/7) Define the required size dimensions, material properties, and the laminate configuration. Discretize the model into finite elements. Consider a tri-linear hexahedron element. Apply any bias factor when discretizing (optional). Calculate the 3D elasticity matrix, . i.e. Compute the elemental matrices necessary in developing the stiffness equation. Specify the boundary conditions (includes the external load).
Bias Factor: Allows us to have more concentrated mesh density near a particular region of interest.
The spacing between the node becomes a geometrical series.
C. Kaufmann, N. Putnam, E. Seo, J. Shin
0 100 200 300 400 5000
20
40
60
80
100Node Profile
x-axis
y-ax
isFiner m
esh
11
Pre-processing (5/7) Define the required size dimensions, material properties, and the laminate configuration. Discretize the model into finite elements. Consider a tri-linear hexahedron element. Apply any bias factor when discretizing (optional). Calculate the 3D elasticity matrix, . i.e. Compute the elemental matrices necessary in developing the stiffness equation. Specify the boundary conditions (includes the external load).
The three-dimensional elasticity (stiffness) matrix is defined by applying the generalized Hooke’s Law.
Take into account the anisotropy, assuming a transversely isotropic layer.
Material nonlinearity (plasticity) is neglected!
C. Kaufmann, N. Putnam, E. Seo, J. Shin
{𝜎 }=[𝑻 1 (−𝜃 ) ] [𝑪 ] [𝑻 2 (𝜃 ) ] {𝜀 }
{𝜎 }=[𝜎 𝑥𝑥
𝜎 𝑦𝑦
𝜎 𝑧𝑧
𝜏 𝑦𝑧𝜏 𝑥𝑧𝜏𝑥𝑦
] {𝜀 }=[𝜀𝑥𝑥𝜀𝑦𝑦𝜀𝑧𝑧𝛾𝑦𝑧𝛾𝑥𝑧𝛾𝑥𝑦
][𝑪 ]=[
1−𝜈23𝜈32𝐸2𝐸3 Δ
𝜈21+𝜈23𝜈31𝐸2𝐸3 Δ
𝜈31+𝜈21𝜈32𝐸2𝐸3 Δ
0 0 0
𝜈21+𝜈23𝜈31𝐸2𝐸3 Δ
1−𝜈13𝜈31𝐸1𝐸3 Δ
𝜈32+𝜈12𝜈31𝐸1𝐸3 Δ
0 0 0
𝜈31+𝜈21𝜈32𝐸2𝐸3 Δ
𝜈32+𝜈12𝜈31𝐸1𝐸3 Δ
1−𝜈12𝜈21𝐸1𝐸2 Δ
0 0 0
0 0 0 𝐺23 0 00 0 0 0 𝐺13 00 0 0 0 0 𝐺12
]Δ≡
1−𝜈12𝜈21−𝜈23𝜈32−𝜈13𝜈31−2𝜈21𝜈32𝜈13𝐸1𝐸2𝐸3
12
Pre-processing (6/7) Define the required size dimensions, material properties, and the laminate configuration. Discretize the model into finite elements. Consider a tri-linear hexahedron element. Apply any bias factor when discretizing (optional). Calculate the 3D elasticity matrix, . i.e. Compute the elemental matrices necessary in developing the stiffness equation. Specify the boundary conditions (includes the external load).
C. Kaufmann, N. Putnam, E. Seo, J. Shin
[𝑵𝑒 ]=[𝑁1𝑒 0 0 𝑁2
𝑒 0 0 ⋯ 𝑁nen𝑒 0 0
0 𝑁 1𝑒 0 0 𝑁2
𝑒 0 ⋯ 0 𝑁nen𝑒 0
0 0 𝑁1𝑒 0 0 𝑁 2
𝑒 ⋯ 0 0 𝑁 nen𝑒 ]
[𝑩𝑒 ]≡ [𝛁s𝑵𝑒 ]=[
𝜕𝑁 1𝑒
𝜕 𝑥0 0
𝜕𝑁 2𝑒
𝜕 𝑥0 0 ⋯
𝜕𝑁 nen𝑒
𝜕 𝑥0 0
0𝜕𝑁 1
𝑒
𝜕 𝑦0 0
𝜕𝑁2𝑒
𝜕 𝑦0 ⋯ 0
𝜕 𝑁nen𝑒
𝜕 𝑦0
0 0𝜕𝑁 1
𝑒
𝜕 𝑧0 0
𝜕𝑁2𝑒
𝜕 𝑧⋯ 0 0
𝜕𝑁nen𝑒
𝜕 𝑧
0𝜕𝑁 1
𝑒
𝜕 𝑧𝜕𝑁 1
𝑒
𝜕 𝑦0
𝜕𝑁2𝑒
𝜕 𝑧𝜕𝑁2
𝑒
𝜕 𝑦⋯ 0
𝜕 𝑁nen𝑒
𝜕𝑧𝜕𝑁nen
𝑒
𝜕 𝑦𝜕𝑁 1
𝑒
𝜕 𝑧0
𝜕𝑁 1𝑒
𝜕 𝑥𝜕𝑁 2
𝑒
𝜕 𝑧0
𝜕𝑁2𝑒
𝜕 𝑥⋯
𝜕𝑁 nen𝑒
𝜕 𝑧0
𝜕𝑁nen𝑒
𝜕 𝑥𝜕𝑁 1
𝑒
𝜕 𝑦𝜕𝑁 1
𝑒
𝜕 𝑥0
𝜕𝑁 2𝑒
𝜕 𝑦𝜕𝑁2
𝑒
𝜕 𝑥0 ⋯
𝜕𝑁 nen𝑒
𝜕 𝑦𝜕 𝑁nen
𝑒
𝜕𝑥0
]𝑁 𝑖
𝑒=𝐿𝐼𝑒 (𝜉 ) 𝐿 𝐽
𝑒 (𝜂 )𝐿𝐾𝑒 (𝜁 )
𝐿𝑚𝑒 (𝜉 )=∏
𝑗 ≠𝑚
𝑝+1 𝜉−𝜉 𝑗𝑒
𝜉𝑚𝑒 −𝜉 𝑗
𝑒 𝐿𝑚𝑒 (𝜂 )=∏
𝑗 ≠𝑚
𝑝+1 𝜂−𝜂 𝑗𝑒
𝜂𝑚𝑒 −𝜂 𝑗
𝑒 𝐿𝑚𝑒 (𝜁 )=∏
𝑗≠𝑚
𝑝+1 𝜁 −𝜁 𝑗𝑒
𝜁𝑚𝑒 −𝜁 𝑗
𝑒
d 𝐿𝑚𝑒 (𝜉 )d 𝜉
=∑h≠𝑚
𝑝+11
𝜉𝑚𝑒 −𝜉 h
𝑒 ( ∏𝑗≠ h∧ 𝑗≠𝑚
𝑝+1 𝜉−𝜉 𝑗𝑒
𝜉𝑚𝑒 −𝜉 𝑗
𝑒 )d 𝐿𝑚
𝑒 (𝜂 )d 𝜂
=∑h≠𝑚
𝑝+11
𝜂𝑚𝑒 −𝜂h
𝑒 ( ∏𝑗 ≠h∧ 𝑗≠𝑚
𝑝+1 𝜂−𝜂 𝑗𝑒
𝜂𝑚𝑒 −𝜂 𝑗
𝑒 )d 𝐿𝑚
𝑒 (𝜁 )d 𝜁
=∑h≠𝑚
𝑝+11
𝜁𝑚𝑒 −𝜁h
𝑒 ( ∏𝑗 ≠h∧ 𝑗≠𝑚
𝑝+1 𝜁 −𝜁 𝑗𝑒
𝜁𝑚𝑒 −𝜁 𝑗
𝑒 )
13
Pre-processing (7/7) Define the required size dimensions, material properties, and the laminate configuration. Discretize the model into finite elements. Consider a tri-linear hexahedron element. Apply any bias factor when discretizing (optional). Calculate the 3D elasticity matrix, . i.e. Compute the elemental matrices necessary in developing the stiffness equation. Specify the boundary conditions (includes the external load).
Essential Boundary Condition
Natural Boundary Condition Pressure-based load, , at
C. Kaufmann, N. Putnam, E. Seo, J. Shin
14
Processing (1/3) “Globalize” and assemble the local stiffness matrices and the local load vectors. Partition and rearrange the global stiffness equation. Solve for the nodal displacement field!
The weak form of our finite element formulation is given below.
Use Gauss Quadrature rule to numerically evaluate the local integration.
Apply the transformation rule to and using the connectivity matrix.
Sum individual matrices and vectors for global assembly.
C. Kaufmann, N. Putnam, E. Seo, J. Shin
∑𝑒=1
nel (𝑳𝑒⊤∫−1
+1
∫−1
+1
∫− 1
+1
𝑩𝑒⊤𝑫𝑒𝑩𝑒| 𝑱𝑒|d 𝜉 d𝜂 d 𝜁 𝑳𝑒)⏟𝑲
𝑑=∑𝑒=1
nel
(𝑳𝑒⊤∫Γ t𝑒
𝑵 𝑒⊤ 𝑡𝑒dΓ )⏟𝑓
15
Processing (2/3) “Globalize” and assemble the local stiffness matrices and the local load vectors. Partition and rearrange the global stiffness equation. Solve for the nodal displacement field!
Partition the global stiffness equation into the known and unknown components.
Apply the transformation rule to rearrange them as shown below.
C. Kaufmann, N. Putnam, E. Seo, J. Shin
[𝑲 ] {𝑑 }= { 𝑓 }
[ 𝑲𝐸 𝑲 𝐸𝐹
𝑲 𝐸𝐹⊤ 𝑲 𝐹
] [𝑑𝐸𝑑𝐹 ]=[ 𝑓 𝐸𝑓 𝐹 ]
16
Processing (3/3) “Globalize” and assemble the local stiffness matrices and the local load vectors. Partition and rearrange the global stiffness equation. Solve for the nodal displacement field!
Solve the system of equations efficiently by using the Gaussian elimination method. In MATLAB, a built-in function, d=K\f can be employed.
C. Kaufmann, N. Putnam, E. Seo, J. Shin
𝑲 𝐸𝐹⊤ 𝑑𝐸+𝑲 𝐹 𝑑𝐹= 𝑓 𝐹
𝑑𝐹=𝑲 𝐹−1 𝑓 𝐹 , ∀𝑑𝐸= 0⃑
𝑓 𝐸=𝑲 𝐸𝐹 𝑑𝐹=𝑲 𝐸𝐹 𝑲 𝐹−1 𝑓 𝐹
17
Post-processing (1/2) Compute the strain field by applying the kinematic equation (displacement ↔ strain). Compute the stress field by applying the constitutive equation (Hooke’s Law).
The elemental strain vector can be computed as shown below. Weighted average of the strain values, evaluated the Gauss points.
C. Kaufmann, N. Putnam, E. Seo, J. Shin
𝜀=𝛁s𝑢
[𝜀𝑥𝑥𝜀𝑦𝑦𝜀𝑧𝑧𝛾𝑦𝑧𝛾𝑥𝑧𝛾𝑥𝑦
]=[𝜕𝜕 𝑥
0 0
0𝜕𝜕 𝑦
0
0 0𝜕𝜕 𝑧
0𝜕𝜕 𝑧
𝜕𝜕 𝑦
𝜕𝜕 𝑧
0𝜕𝜕 𝑥
𝜕𝜕 𝑦
𝜕𝜕 𝑥
0
] [𝑢𝑥𝑢𝑦𝑢𝑧 ]
18
Post-processing (2/2) Compute the strain field by applying the kinematic equation (displacement ↔ strain). Compute the stress field by applying the constitutive equation (Hooke’s Law).
The elemental stress vector can be computed as shown below. Weighted average of the stress values, evaluated the Gauss points.
C. Kaufmann, N. Putnam, E. Seo, J. Shin
{𝜎 }=[𝑫 ] {𝜀 }=[𝑻 1 (−𝜃 ) ] [𝑪 ] [𝑻 2 (𝜃 ) ] {𝜀 }
𝜎 vm=1
√2 √ (𝜎𝑥𝑥−𝜎 𝑦𝑦 )2+(𝜎 𝑦𝑦−𝜎 𝑧𝑧 )2+(𝜎 𝑥𝑥−𝜎 𝑧𝑧 )2+6 (𝜏 𝑦𝑧2 +𝜏𝑥𝑧2 +𝜏 𝑥𝑦
2 )
𝜆3− 𝐼1 𝜆2− 𝐼 2 𝜆− 𝐼3=0
det (𝝈− 𝜆 𝑰 )=0
𝐼 1=𝜎𝑥𝑥+𝜎 𝑦𝑦+𝜎 𝑧𝑧
𝐼 2=𝜎 𝑦𝑦𝜎 𝑧𝑧+𝜎𝑥𝑥𝜎𝑧𝑧+𝜎𝑥𝑥𝜎 𝑦𝑦−𝜏 𝑦𝑧2 −𝜏𝑥𝑧
2 −𝜏 𝑥𝑦2
𝐼 3=𝜎 𝑥𝑥𝜎 𝑦𝑦𝜎𝑧𝑧−𝜎𝑥𝑥𝜏 𝑦𝑧2 −𝜎 𝑦𝑦 𝜏𝑥𝑧
2 −𝜎 𝑧𝑧 𝜏𝑥𝑦2 +2𝜏 𝑦𝑧𝜏𝑥𝑧𝜏𝑥𝑦
𝜏max=MAX (|𝜎 p2−𝜎 p 3
2 |,|𝜎 p1−𝜎 p 3
2 |,|𝜎 p1−𝜎 p 2
2 |)
𝜆1=𝜎 p1 𝜆2=𝜎 p2 𝜆3=𝜎 p3
19
Pathwise-results!C. Kaufmann, N. Putnam, E. Seo, J. Shin
These results refer to a angle-ply laminate.
16800 seconds ≈ 4.5 hours!
0 5 10 15 20 25 30 35
-1
-0.5
0
0.5
1x 106 Stress_yz for [-35 35]
s laminate
Along z-direction [m]
yz [
Pa]
Ply interface
𝜏 𝑦𝑧0 5 10 15 20 25 30 35
-3
-2.5
-2
-1.5
-1
-0.5
x 107 Stress_xz for [-35 35]s laminate
Along z-direction [m]
xz [
Pa]
Stress singularity!
𝜏𝑥𝑧0 20 40 60 80 100
-18000
-16000
-14000
-12000
-10000
-8000
-6000
-4000
-2000
Stress_yz for [-35 35]s laminate
Along y-direction [m]
yz [
Pa]
𝜏 𝑦𝑧
20
Contour Plots!
𝝈𝒙𝒙 𝝈𝒚𝒚 𝝈𝒛𝒛
𝝉𝒚𝒛 𝝉𝒙𝒛 𝝉𝒙𝒚
These results refer to a angle-ply laminate.
C. Kaufmann, N. Putnam, E. Seo, J. Shin
21
More contours…C. Kaufmann, N. Putnam, E. Seo, J. Shin
𝝈𝐩𝟑𝝈𝐩𝟐𝝈𝐩𝟏
𝝈𝐯𝐦 𝝉𝐦𝐚𝐱
These results refer to a angle-ply laminate.
22
Sanity Check! A force reaction balance check would indicate that our analysis was modeled correctly!
External load Reaction load≟• Reaction load is extracted from nodes (DOF to be more specific), where an essential BC is specified.
C. Kaufmann, N. Putnam, E. Seo, J. Shin
Input
Laminate width, Laminate thickness, Pressure Load,
𝑓 𝐸=𝑲𝐸𝐹 𝑑𝐹
𝑲 𝐸𝑑𝐸+𝑲𝐸𝐹 𝑑𝐹= 𝑓 𝐸
Nodal Reaction Forces,
External Pressure Load,
∑𝐹 𝑥= 𝑓 ext+ 𝑓 r=0
23
Error Analysis• and Energy error norms are considered.
• In the below formula, is equal to 1, since a linear element is considered.
C. Kaufmann, N. Putnam, E. Seo, J. Shin
||𝑒||L2=||𝑢 (𝑥 )−𝑢h (𝑥 )||L2=(∑∫𝑥1𝑒
𝑥2𝑒
(𝑢 (𝑥 )−𝑢h (𝑥 ) )2d 𝑥)12
||𝑒||en=||𝑢 (𝑥 )−𝑢h (𝑥 )||en=(∑ 12∫𝑥1𝑒
𝑥2𝑒
𝐸𝑒 (𝜀 (𝑥 )−𝜀h (𝑥 ) )2d 𝑥)12
||𝑒||L2≤𝐶 h𝑝+1 ||𝑒||en ≤𝐶h𝑝
h=√h𝑥2+h𝑦2+h𝑧2
24
Comparison to Ansys?C. Kaufmann, N. Putnam, E. Seo, J. Shin
5.3539e7 Pa(Ansys ACP)
5.358e7 Pa(MATLAB)
𝜎 𝑥𝑥
Averaged over an element