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wiki/concepts/Solid Element Strain-Displacement Matrix.md

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+---
+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]]