we have shown that the material derivative of the jacobian of the deformation gradient tensor can be determined by d/dt(j(y,t)) through indicial notation.
In order to show that the material derivative of the jacobian of the deformation gradient tensor can be determined by d/dt(j(y,t)) through indicial notation, let us start by first defining some of the notations used in this problem. Definitions: Jacobian of a deformation gradient tensor: The Jacobian of a deformation gradient tensor is defined as the determinant of the deformation gradient tensor. It is denoted by J(y,t).Material Derivative: The material derivative of a given quantity, represented by f(y,t), is defined as:df(y,t)/dt = (∂f/∂t) + (v·∇)fwhere v is the velocity of the fluid (or in this case, the material)Gradient: The gradient of a given scalar function, represented by f(y,t), is defined as the vector of its partial derivatives with respect to its independent variables. It is denoted by ∇f.Indicial Notation: Indicial Notation is a notational method that is used to represent and manipulate vectors, tensors, and other geometrical objects in a concise and unambiguous manner. It is based on the Einstein summation convention, which states that any repeated index in a term of a tensor expression should be summed over all possible values of that index.Indicial Notation is used in this problem to represent the partial derivatives of the deformation gradient tensor with respect to its independent variables, which are the spatial coordinates of the material point being considered.Now, let us apply these definitions and notations to the problem at hand.To begin with, we have the following given information:Jacobian of the deformation gradient tensor = J(y,t)Material Derivative of the Jacobian of the deformation gradient tensor = d/dt(J(y,t))Our task is to show that the material derivative of the Jacobian of the deformation gradient tensor can be determined by d/dt(j(y,t)) through indicial notation.To do this, we will start by expressing the material derivative of J(y,t) using its definition, as follows:df(y,t)/dt = (∂f/∂t) + (v·∇)fwhere f(y,t) = J(y,t)Therefore,df(y,t)/dt = (∂J(y,t)/∂t) + (v·∇)J(y,t)Now, let us use the indicial notation to express the partial derivatives of the deformation gradient tensor with respect to its independent variables. For convenience, we will denote the deformation gradient tensor by F and its partial derivatives by Fi,j, where i and j represent the spatial coordinates of the material point being considered.Thus, we can write the following expression for J(y,t):J(y,t) = det(F) = F1,1F2,2F3,3 - F1,1F2,3F3,2 - F1,2F2,1F3,3 + F1,2F2,3F3,1 + F1,3F2,1F3,2 - F1,3F2,2F3,1Using this expression, we can now use the chain rule of differentiation to find the partial derivatives of J(y,t) with respect to its independent variables. Specifically, we have:∂J(y,t)/∂t = ∂det(F)/∂t = det(F)·tr(F^-1(dF/dt))where tr denotes the trace of a matrix, and F^-1 denotes the inverse of the deformation gradient tensor.Using the indicial notation, we can write this expression as:∂J(y,t)/∂t = J(y,t)·Fi,i^-1·Fi,j·dFj,i/dtwhere we have used the summation convention to sum over the repeated index i.Now, let us look at the second term in the material derivative of J(y,t), which involves the gradient of J(y,t) with respect to its independent variables. Using the expression we derived earlier for J(y,t), we can write:∇J(y,t) = (partial(J)/partial(x1), partial(J)/partial(x2), partial(J)/partial(x3))where x1, x2, and x3 denote the spatial coordinates of the material point being considered.Using the indicial notation, we can write this expression as:∇J(y,t) = (partial(J)/partial(xi))where i = 1,2,3Therefore, the gradient of J(y,t) with respect to its independent variables can be expressed as:∇J(y,t) = J(y,t)·Fi,i^-1·Fi,j·(partial(Fj,k)/partial(xi))Using the chain rule of differentiation, we can express this as:∇J(y,t) = J(y,t)·Fi,i^-1·Fi,j·(dFj,k/dxk)·(dxk/dxi)where we have used the summation convention to sum over the repeated index k.Now, substituting these expressions back into the material derivative of J(y,t), we get:d/dt(J(y,t)) = (∂J(y,t)/∂t) + (v·∇)J(y,t)= J(y,t)·Fi,i^-1·Fi,j·dFj,i/dt + J(y,t)·Fi,i^-1·Fi,j·(dFj,k/dxk)·(dxk/dxi)·vwhich is the desired result.
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