김경종
73 lines
2.2 KiB
Markdown
73 lines
2.2 KiB
Markdown
---
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type: concept
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title: "Solid Element Strain-Displacement Matrix"
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complexity: advanced
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domain: computational-mechanics
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aliases:
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- solid element B matrix
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- 3D strain-displacement matrix
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- Jacobian derivative mapping
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created: 2026-05-28
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updated: 2026-05-28
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address: c-000051
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tags:
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- concept
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- finite-element-method
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- solid-elements
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- strain-displacement
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status: current
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related:
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- "[[Solid Element Notes]]"
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- "[[Displacement-Based Finite Element Formulation]]"
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- "[[Isoparametric Finite Elements]]"
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- "[[Solid Element Shape Functions]]"
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- "[[Solid Element Stiffness Integration]]"
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sources:
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- "[[Solid Element Notes]]"
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source_refs:
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- source: "[[Solid Element Notes]]"
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raw_path: ".raw/SolidElement/"
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raw_files:
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- "SolidElement_001.md"
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md_indices:
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- 1
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match: "heuristic-heading-keyword"
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confidence: low
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---
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# Solid Element Strain-Displacement Matrix
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## Definition
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The solid element strain-displacement matrix, usually called the `B` matrix, maps nodal translational degrees of freedom to the six small-strain components of a three-dimensional continuum element.
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## How It Works
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The notes use the standard small-strain components:
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- normal strains: `epsilon_xx`, `epsilon_yy`, `epsilon_zz`
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- engineering shear terms derived from `epsilon_xy`, `epsilon_yz`, and `epsilon_xz`
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Each node contributes a block of derivatives of its shape function with respect to physical coordinates:
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```text
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[ dN_i/dx 0 0 ]
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[ 0 dN_i/dy 0 ]
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[ 0 0 dN_i/dz ]
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[ dN_i/dy dN_i/dx 0 ]
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[ 0 dN_i/dz dN_i/dy]
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[ dN_i/dz 0 dN_i/dx]
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```
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Because [[Solid Element Shape Functions]] are defined in natural coordinates, their derivatives must be mapped into physical coordinates through the Jacobian. This derivative mapping is the core computational step between interpolation and stiffness assembly.
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## Connections
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- [[Displacement-Based Finite Element Formulation]] supplies the general `epsilon = B u` path.
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- [[Isoparametric Finite Elements]] explains why the Jacobian appears.
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- [[Solid Element Stiffness Integration]] uses this `B` matrix in `K = integral B^T D B dV`.
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## Sources
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- [[Solid Element Notes]]
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