---
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]]"
source_refs:
  - source: "[[Solid Element Notes]]"
    raw_path: ".raw/SolidElement/"
    raw_files:
      - "SolidElement_001.md"
    md_indices:
      - 1
    match: "heuristic-heading-keyword"
    confidence: low
---

# 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]]