김경종
64 lines
2.0 KiB
Markdown
64 lines
2.0 KiB
Markdown
---
|
|
type: concept
|
|
title: "Solid Element Strain-Displacement Matrix"
|
|
complexity: advanced
|
|
domain: computational-mechanics
|
|
aliases:
|
|
- solid element B matrix
|
|
- 3D strain-displacement matrix
|
|
- Jacobian derivative mapping
|
|
created: 2026-05-28
|
|
updated: 2026-05-28
|
|
address: c-000051
|
|
tags:
|
|
- concept
|
|
- finite-element-method
|
|
- solid-elements
|
|
- strain-displacement
|
|
status: current
|
|
related:
|
|
- "[[Solid Element Notes]]"
|
|
- "[[Displacement-Based Finite Element Formulation]]"
|
|
- "[[Isoparametric Finite Elements]]"
|
|
- "[[Solid Element Shape Functions]]"
|
|
- "[[Solid Element Stiffness Integration]]"
|
|
sources:
|
|
- "[[Solid Element Notes]]"
|
|
---
|
|
|
|
# Solid Element Strain-Displacement Matrix
|
|
|
|
## Definition
|
|
|
|
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.
|
|
|
|
## How It Works
|
|
|
|
The notes use the standard small-strain components:
|
|
|
|
- normal strains: `epsilon_xx`, `epsilon_yy`, `epsilon_zz`
|
|
- engineering shear terms derived from `epsilon_xy`, `epsilon_yz`, and `epsilon_xz`
|
|
|
|
Each node contributes a block of derivatives of its shape function with respect to physical coordinates:
|
|
|
|
```text
|
|
[ dN_i/dx 0 0 ]
|
|
[ 0 dN_i/dy 0 ]
|
|
[ 0 0 dN_i/dz ]
|
|
[ dN_i/dy dN_i/dx 0 ]
|
|
[ 0 dN_i/dz dN_i/dy]
|
|
[ dN_i/dz 0 dN_i/dx]
|
|
```
|
|
|
|
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.
|
|
|
|
## Connections
|
|
|
|
- [[Displacement-Based Finite Element Formulation]] supplies the general `epsilon = B u` path.
|
|
- [[Isoparametric Finite Elements]] explains why the Jacobian appears.
|
|
- [[Solid Element Stiffness Integration]] uses this `B` matrix in `K = integral B^T D B dV`.
|
|
|
|
## Sources
|
|
|
|
- [[Solid Element Notes]]
|