A deformation gradient describes how much a body deforms in different directions. Mathematically, it is the deformed vector x taking derivative of each component the reference vector X. So for a 2D case, the deformation gradient is a 2-by-2 matrix, and in 3D case, the deformation gradient is a 3-by-3 matrix.For example, a 2D box (with its edges aligned with x and y axis) is stretched in x direction for 50% and y direction for 200%, then the deformation gradient is [1.5, 0; 0, 3].
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