Extensions of Kronecker product algebra with applications in continuum and computational mechanics

Summary Kronecker product algebra is widely applied in control theory. However, it does not appear to have been commonly applied to continuum and computational mechanics (CCM). In broad terms the goal of the current investigation is to extend Kronecker product algebra so that it can be broadly applied to CCM. Many CCM quantities, such as the tangent compliance tensor in finite strain plasticity, are very elaborate or difficult to derive when expressed in terms of tensor indicial or conventional matrix notation. However, as shown in the current article, with some extensions Kronecker product algebra can be used to derive compact expressions for such quantities. In the following, Kronecker product algebra is reviewed and there are given several extensions, and applications of the extensions are presented in continuum mechanics, computational mechanics and dynamics. In particular, Kronecker counterparts of quadratic products and of tensor outer products are presented. Kronecker operations on block matrices are introduced. Kronecker product algebra is extended to third and fourth order tensors. The tensorial nature of Kronecker products of tensors is established. A compact expression is given for the differential of an isotropic function of a second-order tensor. The extensions are used to derive compact expressions in continuum mechanics, for example the transformation relating the tangent compliance tensor in finite strain plasticity in undeformed to that in deformed coordinates. A compact expression is obtained in the nonlinear finite element method for the tangent stiffness matrix in undeformed coordinates, including the effect of boundary conditions prescribed in the current configuration. The aforementioned differential is used to derive the tangent modulus tensor in hyperelastic materials whose strain energy density is a function of stretch ratios. Finally, block operations are used to derive a simple asymptotic stability criterion for a damped linear mechanical system in which the constituent matrices appear explicitly.Appendix: Notation A, Â, a_ij matrix, second-order tensor - a, a_i vector, VEC (A) - a vector - â scalar - B, b_ij matrix, second-order tensor - b, b_i VEC (B) - b vector - block permutation matrix - C, C, c_ijkl fourth-order tensor - right Cauchy Green strain tensor - c VEC () - c_i eigenvalues of - C_a,C_b third-order tensors - C₁,C₂,C₃ outer product functions - D deformation rate tensor - D damping matrix - d VEC (D) - E boundary stiffness matrix - e VEC () - Eulerian strain - F isotropic tensor-valued function ofA - deformation gradient tensor - f VEC - f scalar valued counterpart ofF - G coordinate transformation tensor - G strain-displacement matrix - g VEC (G) - g consistent force vector - H dynamic system matrix - h_n lowest eigenvalue ofH - I,I_n,I₉ identity matrix/tensor - i VEC (I) - I₁ TRACE () - I index - i index - J determinant of - J matrix relating d to da - J index - j index - J matrix relating d to da - K,K_T,K_b,K stiffness matrices - K index - k index - L velocity gradient tensor - L index - l index - M mass matrix - strain-displacement matrix - M matrix arising from Ogden model - M index - m index - N shape function matrix - N matrix arising from Ogden model - n, n₀ exterior normal vectors - n n² - N index - n index - P, dynamic system matrix - p VEC (P) - p_n eigenvalue ofP - Q rotation tensor - R,r_ij tensor used with outer products - r rank, index - r VEC (R) - S,s_ij tensor used with outer products - s VEC (S) - S, S₀ surface area - S matrix diagonalizingA - s index - T unitary matrix - t₀, t traction - t VEC () - VEC () - t time - T time interval - U_n,U₉,U_M permutation matrices - u displacement vector - V matrix appearing in linear system stability criterion - V projection matrix - W,W_i multipliers d - w, w₁ vectors - strain energy function - X undeformed position - x deformed position - Y,Y coordinate system - y, y_j vectors - z, z_j vectors - _j eigenvalue ofA - _j eigenvalue ofB - , ₁, ₂ nodal displacement vectors - _j eigenvalue of - matrix - , ₁, ₂, _a, _b diagonal matrices - _ij entries of the Kronecker tensor (I) - Lagrangian strain - _ijk permutation tensor - _i coefficient of Ogden model - parameter in linear system stability criterion - _i eigenvalue - Lamé coefficient - matrix - surface area factor - Lamé coefficient - _j eigenvalue ofA - _i coefficient of Ogden model - matrix - second Piola-Kirchhoff stress - Cauchy stress - Truesdell stress flux - matrix/tensor - matrix/tensor - matrix/tensor - TEN22 (C) - d rotation vector - d rotation tensor - TRACE(.) trace of a matrix - left Kronecker function - right Kronecker function - VEC(.) vectorization operator - VECB(.) block vectorization operator - TEN22(.) tensor operator - TEN12(.) tensor operator - TEN21(.) tensor operator - _x, _y divergence operator - d(.) differential operator - (.) variational operator - (right) Kronecker product - Kronecker sum - Kronecker difference - block Kronecker product - left Kronecker product - AsB block Kronecker sum ofA andB - AdB block Kronecker difference ofA and