Tangent stiffness matrix

About: Tangent stiffness matrix is a research topic. Over the lifetime, 1031 publications have been published within this topic receiving 21140 citations.

Numerical investigation of alternative fracture stiffness measures and their respective scaling behaviours

Abstract: We study the mechanical deformation of fractures under normal stress, via tangent and specific fracture stiffnesses, for different length scales using numerical simulations and analytical insights. First, we revisit an equivalent elastic layer model that leads to two expressions: the tangent stiffness is the sum of an “intrinsic” stiffness and the normal stress, and the specific stiffness is the tangent stiffness divided by the fracture aperture at current stress. Second, we simulate the deformation of rough fractures using a boundary element method where fracture surfaces represented by elastic asperities on an elastic half-space follow a self-affine distribution. A large number of statistically identical “parent” fractures are generated, from which sub-fractures of smaller dimensions are extracted. The self-affine distribution implies that the stressfree fracture aperture increases with fracture length with a power law in agreement with the chosen Hurst exponent. All simulated fractures exhibit an increase in the specific stiffness with stress and an average decrease with increase in length consistent with field observations. The simulated specific and tangent stiffnesses are well described by the equivalent layer model provided the “intrinsic” stiffness slightly decreases with fracture length following a power law. By combining numerical simulations and the analytical model, the effect of scale and stress on fracture stiffness measures can be easily separated using the concept of “intrinsic” stiffness. We learn that the primary reason for the variability in specific stiffness with length comes from the fact that the typical aperture of the self-affine fractures itself scales with the length of the fractures.

Characteristics of the Solution of the Consistently Linearized Eigenproblem for Lateral Torsional Buckling

Abstract: The consistently linearized eigenproblem has proved to be a powerful mathematical tool for classification of buckling, based on the percentage bending energy of the total strain energy. Of particular interest are prebuckling states with a constant percentage strain energy. The two limiting cases of such states are membrane stress states and states of pure bending. Buckling at pure bending, referred to as lateral torsional buckling, is the topic of this work. The transfer matrix method is used to derive a secant stiffness matrix in analytical form. Formulation of the consistently linearized eigenproblem by means of this matrix yields the same solution as would be obtained by a formulation based on the tangent stiffness matrix which is an essential ingredient of nonlinear Finite Element Analysis. This remarkable finding permits analytical verification of hypothesized subsidiary conditions for lateral torsional buckling. © 2012 Bull. Georg. Natl. Acad. Sci.

Adaptive time stepping in pseudo-dynamic testing of earthquake driven structures

Abstract: The problem of assessing errors in implementing time-marching algorithms in the context of pseudo-dynamic seismic testing of structures is considered. These errors occur in implementing the numerical and experimental steps of the test procedure. The study investigates how a linearized variational equation can be augmented with the governing equation of motion to track the effect of the errors, and, accordingly, adjust the step size of integration adaptively to keep a global error norm within specified limits. The governing augmented equations are integrated using an explicit operator splitting scheme. Additional efforts, in terms of evaluation of the tangent stiffness matrix, are shown to become necessary while modelling the errors. Illustrative examples include numerical studies on a set of nonlinear systems and an experimental study on a geometrically nonlinear two-storied building frame. The experimental results from pseudo-dynamic test are shown to compare reasonably well with pertinent results from an effective force test.

Implicit Integration and Consistent Tangent Modulus of a Time-Dependent Non-Unified Constitutive Model.

Abstract: This paper is concerned with the implicit integration and consistent tangent modulus of a high-temperature constitutive model, in which time-dependent inelastic strain rate consists of the transient part affected by kinematic and isotropic hardenings and the steady part depending on stress and temperature. Such a model is useful for high-temperature structure analysis and is practical because of the ease in determining material constants. The implicit integration is shown to result in two scalar-valued equations, and the consistent tangent modulus is derived in a general form by introducing a set of fourth-rank constitutive parameters into discretized kinematic hardening. The constitutive model is, then, implemented in a finite element program and applied to lead-free solder joint analysis. It is demonstrated that the implicit integration is very accurate if the kinematic hardening model of Ohno and Wang is employed, and that the consistent tangent modulus affords parabolic convergence to the Newton-Raphson iteration for solving nodal force equilibrium equations.