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Is the deformation gradient symmetric?

Is the deformation gradient symmetric?

This is reflected in the deformation gradient by the fact that it is not symmetric.

What is linear elastic deformation?

Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics.

What is determinant of deformation gradient?

The determinant of the deformation gradient is usually denoted by J {\displaystyle J} and is a measure of the change in volume, i.e., J = det F {\displaystyle J=\det {\boldsymbol {F}}}

What is a displacement gradient?

In this book quantities like ∂u/∂x, the gradients of the displacement field, are called displacement gradients, and quantities like ∂X/∂x, the gradients of the coordinate transformation field, are called deformation gradients.

What is pure deformation?

For an infinitesimal fibre that deforms from an initial point given by the vector dX to the vector dx in the time t, the deformation gradient is defined by Fij = ∂xi(X, t)/∂Xj; the 3 × 3 matrix [F], with components Fij, may be represented as a pure deformation, characterized by a symmetric matrix [U], followed by a …

What is considered a large deformation?

In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strain theory.

What is Green’s strain?

Green Strain = Small Strain Terms + Quadratic Terms. The small strain terms are the same, possessing all the desirable properties of engineering strain behavior. The quadratic terms are what gives the Green strain tensor its rotation independence.

Is displacement gradient symmetric?

A simple shear deformation is always accompanied by a rigid-body rotation, because the displacement gradient tensor is not symmetrical.

What is the gradient of a velocity?

The difference in velocity between adjacent layers of the fluid is known as a velocity gradient and is given by v/x, where v is the velocity difference and x is the distance between the layers.

How is the deformation gradient used in physics?

The deformation gradient is used to separate rigid body translations and rotations from deformations, which are the source of stresses.

Why do rigid body displacements not appear in the deformation gradient?

Clearly rigid body displacements do not appear in the deformation gradient. This is good because rigid body displacements don’t contribute to stress, strain, etc. These equations rotate an object counter-clockwise about the z z axis.

How to calculate the deformation gradient in uniaxial compression?

At first sight these two deformation gradients look quite different but if we select λ = 0.5 in uniaxial compression, and λ = 2 in biaxial tension, then the deformation gradient in both cases becomes (5.58) F = [0.5 0 0 0 2 0 0 0 2].

Why are deformation gradients not detected in FE analysis?

This effect is known as hourglassing in FE analyses and arises in 2-D quadrilateral and 3-D brick elements undergoing reduced integration. In such cases, element deformations are only evaluated at the center of the element. As a result, hourglass deformations are not detected and can grow uncontrollably.