For the motion x1 = αξ1 − βξ1ξ2, x2 = βξ1 + αξ2, x3 = ξ3 (a) Determine the components of the deformation gradient F. (b) Find the stretch of a line element located at (0, 0, 0) along the direction M = (e1 + e2) / √ 2, where M is the unit vector in the initial material (Lagrangian) configuration. (c) Determine the total volume of the body in the current configuration.

To determine the components of the deformation gradient F, we can calculate the partial derivatives of the given motion equations with respect to the initial coordinates ξ1, ξ2, and ξ3. Given: x1 = αξ1 - βξ1ξ2 x2 = βξ1 + αξ2 x3 = ξ3 (a) Components of the deformation gradient F: The deformation gradient F is a matrix that represents the local deformation of a material. It is defined as the derivative of the current position vector x with respect to the initial position vector ξ. F = ∂x/∂ξ Taking the partial derivatives, we have: ∂x1/∂ξ1 = α - βξ2 ∂x1/∂ξ2 = -βξ1 ∂x1/∂ξ3 = 0 ∂x2/∂ξ1 = β ∂x2/∂ξ2 = α ∂x2/∂ξ3 = 0 ∂x3/∂ξ1 = 0 ∂x3/∂ξ2 = 0 ∂x3/∂ξ3 = 1 Therefore, the deformation gradient F is given by: F = [[∂x1/∂ξ1, ∂x1/∂ξ2, ∂x1/∂ξ3], [∂x2/∂ξ1, ∂x2/∂ξ2, ∂x2/∂ξ3], [∂x3/∂ξ1, ∂x3/∂ξ2, ∂x3/∂ξ3]] Substituting the partial derivatives, we get: F = [[α - βξ2, -βξ1, 0], [β, α, 0], [0, 0, 1]] (b) Stretch of a line element along the direction M: To find the stretch of a line element, we need to calculate the length of the line element in the current configuration and divide it by its length in the initial configuration. In the initial configuration, the line element is located at (0, 0, 0) and has a direction M = (e1 + e2) / √2, where e1 and e2 are the basis vectors. The length of the line element in the initial configuration is L0 = √(ξ1^2 + ξ2^2 + ξ3^2) = 0 (since it's located at the origin). To find the length of the line element in the current configuration, we apply the deformation gradient F to the direction vector M. M' = F * M Substituting the values, we have: M' = [[α - βξ2, -βξ1, 0], [β, α, 0], [0, 0, 1]] * [(1/√2), (1/√2), 0] Simplifying the matrix multiplication, we get: M' = [(α - βξ2)/√2, (-βξ1 + α)/√2, 0] The length of the line element in the current configuration is L = √((α - βξ2)^2/2 + (-βξ1 + α)^2/2 + 0^2) = √((α - βξ2)^2 + (-βξ1 + α)^2)/√2 Therefore, the stretch of the line element along the direction M is given by: Stretch = L / L0 = √((α - βξ2)^2 + (-βξ1 + α)^2)/√2 (c) Total volume of the body in the current configuration: To determine the total volume of the body in the current configuration, we need to calculate the determinant of the deformation gradient F. Det(F) = (α - βξ2)(α)(1) + (-βξ1)(β)(0) + (0)(0)(α) - (0)(α)(α - βξ2) - (α)(-βξ1)(0) - (1)(0)(α - βξ2) Simplifying the expression, we have: Det(F) = α^2 - αβξ2 The total volume of the body in the current configuration is given by the determinant of the deformation gradient. Total volume = Det(F) = α^2 - αβξ2