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

type, title, complexity, domain, aliases, created, updated, address, tags, status, related, sources

type	title	complexity	domain	aliases	created	updated	address	tags	status	related	sources
concept	Solid Element Strain-Displacement Matrix	advanced	computational-mechanics	solid element B matrix 3D strain-displacement matrix Jacobian derivative mapping	2026-05-28	2026-05-28	c-000051	concept finite-element-method solid-elements strain-displacement	current	Solid Element Notes Displacement-Based Finite Element Formulation Isoparametric Finite Elements Solid Element Shape Functions Solid Element Stiffness Integration	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:

[ 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