Stiffness Matrix
The stiffness matrix [C] relates stress σ to strain ε: σ = [C]ε. For isotropic materials, only two constants (E, ν or λ, μ). 3D: 6×6; plane stress/strain: 3×3.
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Isotropic: 2 constants (E, ν) or (λ, μ). Plane stress: thin plates, out-of-plane σ=0. Plane strain: long structures, ε_z=0. Coordinate transform for rotated axes.
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Why: Stiffness matrix is fundamental in FEM, stress analysis, and continuum mechanics. Isotropic materials simplify to 2 independent constants.
How: From E and ν compute Lamé λ, μ. Build 6×6 matrix for 3D. Plane stress/strain reduce to 3×3. Rotation transforms via T^T C T.
Run the calculator when you are ready.
🔩 Steel 3D Matrix
Steel: E=200 GPa, ν=0.29 - Full 6x6 stiffness matrix
⚙️ Aluminum Plane Stress
Aluminum plane stress stiffness matrix (3x3)
📐 Aluminum Plane Strain
Aluminum plane strain stiffness matrix (3x3)
🔧 Custom E and ν
Calculate from E=100 GPa, ν=0.3
🔄 Coordinate Rotation
Steel with 45° rotation about z-axis
Enter Values
Calculation Mode
Analysis Type
Material Selection
Coordinate Transformation
Elastic Constants
Young's Modulus
Poisson's Ratio
Shear Modulus
Bulk Modulus
Lamé Parameters
3D Stiffness Matrix [C] (6×6)
| εxx | εyy | εzz | γxy | γyz | γzx | |
| σxx | 262.089 GPa | 107.051 GPa | 107.051 GPa | 0 GPa | 0 GPa | 0 GPa |
| σyy | 107.051 GPa | 262.089 GPa | 107.051 GPa | 0 GPa | 0 GPa | 0 GPa |
| σzz | 107.051 GPa | 107.051 GPa | 262.089 GPa | 0 GPa | 0 GPa | 0 GPa |
| τxy | 0 GPa | 0 GPa | 0 GPa | 77.519 GPa | 0 GPa | 0 GPa |
| τyz | 0 GPa | 0 GPa | 0 GPa | 0 GPa | 77.519 GPa | 0 GPa |
| τzx | 0 GPa | 0 GPa | 0 GPa | 0 GPa | 0 GPa | 77.519 GPa |
Matrix Properties
Relationships
C₁₁ from E and ν: C₁₁ = λ + 2μ = 261.45 GPa
C₁₂ from E and ν: C₁₂ = λ = 107.05 GPa
C₄₄ from G: C₄₄ = μ = G = 77.20 GPa
E from C: E = C₁₁ - 2C₁₂ = 47.35 GPa
G from C: G = C₄₄ = 77.20 GPa
K from C: K = (C₁₁ + 2C₁₂)/3 = 158.52 GPa
Visualizations
Material Comparison
Recommendations
Good stiffness for structural applications
Step-by-Step Calculation
Material: Steel (Carbon)
Young's Modulus (E): 200 GPa
Poisson's Ratio (ν): 0.29
Shear Modulus (G): 77.2 GPa
Bulk Modulus (K): 160 GPa
λ = Eν / ((1+ν)(1-2ν)) = 107.05 GPa
μ = G = 77.20 GPa
3D Stiffness Matrix [C] (6×6)
C₁₁ = λ + 2μ = 107.05 GPa + 2×77.20 GPa = 261.45 GPa
C₁₂ = λ = 107.05 GPa
C₄₄ = μ = G = 77.20 GPa
For educational and informational purposes only. Verify with a qualified professional.
🔬 Physics Facts
σ = [C]ε; stress-strain relation
— Elasticity
Isotropic: C₁₁=λ+2μ, C₁₂=λ, C₄₄=μ
— Continuum mechanics
λ, μ = Lamé parameters from E, ν
— Elasticity
C' = T^T C T for rotation
— Tensor transformation
What is a Stiffness Matrix?
The stiffness matrix [C] is a fundamental tensor in continuum mechanics that relates stress to strain in linear elastic materials. For isotropic materials, the stiffness matrix has a special symmetric form that depends on only two independent elastic constants (typically Young's modulus E and Poisson's ratio ν). The relationship is expressed as σ = [C]ε, where σ is the stress vector and ε is the strain vector.
3D Stiffness Matrix
6×6 matrix relating all stress and strain components. Used for full 3D analysis.
Plane Stress
3×3 matrix for thin plates where σzz = τxz = τyz = 0. Assumes zero out-of-plane stress.
Plane Strain
3×3 matrix for thick structures where εzz = γxz = γyz = 0. Assumes zero out-of-plane strain.
Matrix Components and Relationships
3D Isotropic Stiffness Matrix
C₁₂ = λ = Eν/((1+ν)(1-2ν))
C₄₄ = μ = G = E/(2(1+ν))
Where λ is the first Lamé parameter and μ (shear modulus) is the second Lamé parameter.
Plane Stress Matrix
C₁₂ = νE/(1-ν²)
C₃₃ = G = E/(2(1+ν))
Used for thin plates, membranes, and shells where thickness is much smaller than in-plane dimensions.
Plane Strain Matrix
C₁₂ = λ
C₃₃ = μ = G
Used for thick plates, long structures, and problems where out-of-plane deformation is constrained.
Coordinate Transformations
When analyzing materials at different orientations, the stiffness matrix must be transformed. For a rotation about an axis, the transformed stiffness matrix is calculated as C' = T^T C T, where T is the transformation matrix. This is essential for analyzing anisotropic materials or rotated coordinate systems.
Frequently Asked Questions
Q1: What is the difference between stiffness matrix and compliance matrix?
The stiffness matrix [C] relates stress to strain (σ = [C]ε), while the compliance matrix [S] is its inverse and relates strain to stress (ε = [S]σ). For isotropic materials, they are inverses of each other.
Q2: When should I use plane stress vs plane strain?
Plane stress applies to thin plates where out-of-plane stress is zero (σzz = 0). Plane strain applies to thick structures where out-of-plane strain is zero (εzz = 0). Choose based on your geometry and loading conditions.
Q3: What do the Lamé parameters represent?
The first Lamé parameter (λ) relates to volumetric deformation, while the second (μ) is the shear modulus. They provide an alternative to Young's modulus and Poisson's ratio for describing isotropic elastic behavior.
Q4: How does coordinate transformation affect the stiffness matrix?
Rotating the coordinate system changes the matrix components while preserving material properties. The transformation C' = T^T C T ensures the physical behavior remains unchanged, only the representation changes.
Q5: What does a high condition number indicate?
A high condition number suggests the matrix is near-singular, making numerical inversion unstable. This can occur with extreme Poisson's ratios or when the material approaches incompressibility (ν → 0.5).
Q6: Can I use this calculator for anisotropic materials?
This calculator is designed for isotropic materials. Anisotropic materials require a full 6×6 stiffness matrix with 21 independent constants (reduced from 36 by symmetry). Use specialized FEA software for anisotropic analysis.
Q7: How accurate are these calculations for real materials?
The calculations assume linear elastic behavior and perfect isotropy. Real materials may exhibit nonlinearity, anisotropy, temperature dependence, and rate effects. Always verify with material testing for critical applications.
📚 Official Data Sources
⚠️ Disclaimer
⚠️ Disclaimer: This calculator provides theoretical calculations based on linear elasticity theory for isotropic materials. Actual material behavior may vary significantly due to:
- Nonlinear elastic behavior beyond the elastic limit
- Material anisotropy and texture effects
- Temperature and strain rate dependence
- Manufacturing variations and defects
- Environmental factors (moisture, corrosion, aging)
For critical engineering applications, always verify calculations with experimental testing and consult with qualified materials engineers. This tool is for educational and preliminary design purposes only.
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