F.A. Cozzarelli

Mechanism of subsidence of ancient cratonic rift basins

Cratonic basins commonly occur over ancient rift zones. These inactive rift basins are recognizable by a positive linear Bouguer gravity anomaly that may correspond to the axial gravity high found in modern rift valleys. Many of these basins undergo discrete periods of increased subsidence rates, or reactivations, long after the mass excess responsible for the linear gravity high was emplaced. Furthermore, the reactivation of many cratonic basins occurs simultaneously with large-scale compressional tectonics. It is suggested that the driving force for subsidence is the isostatically uncompensated ancient mass excess. The subsidence of these basins is modelled by a lithospheric flexure model with a nonlinear Maxwell viscoelastic rheology. Solutions to this model indicate that basins may experience a low subsidence rate throughout geologic time. The subsidence of a basin will stop only when isostatic compensation of the mass excess is achieved. Since ancient rift mass excesses may be uncompensated over long geologic time intervals, the early thermal and structural evolution of rifts may not significantly influence later basin subsidence. The models suggest that basins may be reactivated by any mechanism which lowers the effective viscosity of the lithospheric material, allowing the uncompensated basin to settle toward an isostatic-compensation depth faster than normal. Since viscosity is a strong function of temperature, reactivation by a world-wide increase in heat flow is suggested as a possible mechanism for the synchroneity of basin subsidence throughout a continent. An increase of 15% in the geothermal gradient, for example (from 16.5°–18.9°K/km), will cause about a 5% increase in subsidence. This increase in heat flow, however, seems unlikely of producing by itself the magnitude of basin subsidence during a reactivation phase that is observed in the geologic record where up to 100% increase in subsidence might occur. Since the rheology of the lithosphere is nonlinear, effective viscosity is also a strong nonlinear function of stress. The presence of a regional compressive stress during periods of tectonism of 1.1 · 108 Pa (about 2.8% of the buckling strength of the lithosphere) produces a short period of reactivated subsidence (≅ 105 yr). During the reactivated subsidence, the newly-imposed regional stress relaxes sufficiently in the lower lithosphere to restore the effective viscosity to values similar to that before reactivation. This suggests that reactivated subsidence caused by regional compression can be maintained as long as the stress level remains high in the lower lithosphere. This may be accomplished by an intermittent application of the regional stress over time.

Non-linear creep damage under one-dimensional variable tensile stress

Abstract A non-linear damage relation, containing the axial strain history and a time integral over the stress history, is proposed for the case of one-dimensional time dependent tensile stress. Non-linear steady and transient creep terms are included in the axial strain relation, and elastic and creep Poissons ratios are introduced into the lateral strain relation. Using these relations, complete damage solutions are obtained for the constant stress rate, step stress, relaxation and constant load tests. Observations are made concerning the associated rupture times.

EFFECT OF RANDOM MATERIAL PARAMETERS ON NONLINEAR STEADY CREEP SOLUTIONS

Abstract An analysis is presented concerning the effect of random material parameters on nonlinear steady creep in a 3-bar truss. Parameter randomness is in general large and is introduced through randomness in the temperature and in the density of imperfections. Analytical and numerical results are presented on the statistical properties of the material parameters and of the stress and velocity. It is shown that randomness in the material parameters will on the one hand introduce only a very slight randomness in stress and on the other hand a very significant randomness in velocity.

Similarity solutions to some non-linear impact problems

Abstract Impact and wave propagation problems are considered for nonlinearly viscous and nonlinearly elastic materials. The governing partial differential equations are reduced to ordinary differential equations by means of similarity transformations. The resulting non-linear two point boundary value problems are then, in general, integrated numerically, although some closed form solutions are presented.

Stress relaxation at wave fronts in one-dimensional media described by non-linear viscoelastic models

Abstract Constitutive equations are presented for non-linear viscoelastic materials under variable stress loading nd are interpreted in terms of non-linear viscoelastic models. Analytical solutions are obtained for stress, velocity and strain at the wave front in an impulsively loaded semi-infinite rod of material described by these non-linear constitutive equations within the assumptions of small deformation theory.

One-dimensional strain-dependent creep damage in inhomogeneous materials

Abstract A local inhomogeneous strain-dependent creep damage theory is presented and then applied using creep damage data obtained at J.R.C. Ispra for specimens of various lengths. Solutions for one-dimensional constant stress and load are obtained, and are found to be in better agreement with observation than solutions based on previous homogeneous theory. Also, the effect of random material parameters on the rupture time is considered for the constant tensile stress test, and is found to be a significant factor.

THE RANDOM STEADY STATE DIFFUSION PROBLEM. I: RANDOM GENERALIZED SOLUTIONS TO LAPLACE'S EQUATION*

The following three problems for Laplace’s equation are considered: (i) generalized Dirichlet problem for a random boundary function, (ii) generalized Poisson problem for a random right-hand side, and (iii) generalized Dirichlet–Poisson problem for random boundary function and right-hand side. For each of these problems a solution which is an extension of both the deterministic solution and Babuska’s results is defined. Existence, uniqueness, and conditions under which such solutions are equivalent to both sample and mean square solutions are established. In each case the properties of the solution are analyzed.

The Random Steady State Diffusion Problem. III: Solutions to Random Diffusion Problems by the Method of Random Successive Approximations

An iterative method of solving a random integral equation occurring in random steady state diffusion problems is presented. Convergence of such a method which is a stochastic analogue of the deterministic method of successive approximations is established. Properties of the solution so constructed are given. An application to random diffusion in the unit interval is presented. A comparison with a perturbation method is carried out.

Effect of time dependent compressibility on non-linear viscoelastic wave propagation

Abstract The work presented consists essentially of two parts: the first deals with the development of a non-linear constitutive equation for a three-dimensional viscoelastic material with instantaneous and time dependent compressibility; the second deals with the solution of some specific wave propagation problems for three simple three-dimensional geometries. The constitutive equation is based on the existence of elastic and creep potentials and is expressed in terms of single memory integrals with non-linear kernels. The wave propagation problems are solved by numerical integration along the characteristics of the governing equations. The primary conclusion drawn deals with the effect of time dependent compressibility on the dynamic stress, strain and velocity fields. Results indicate that the dynamic response of even slightly time dependent compressible materials varies dramatically from those assumed to have only an instantaneous elastic compressibility.

On the thermodynamics of nonlinear single integral representations for thermoviscoelastic materials with applications to one-dimensional wave propagation

SummaryThermodynamic theory is used to develop single integral constitutive relations for the nonlinear thermoviscoelastic response to arbitrary stress and temperature histories; the thermomechanically coupled energy equation is also obtained. The thermorheologically simple material, modified superposition and the isotropic stress power law are discussed in detail. A modified Fourier heat conduction law is employed to ensure that the propagation of thermal disturbances takes place at a finite velocity. Using the nonlinear thermoviscoelastic stress power law along with the linearized energy equation and modified Fourier law, one-dimensional wave front solutions are obtained.ZusammenfassungMit Hilfe der Thermodynamik werden einfache Integrale enthaltende Werkstoffbeziehungen für das nichtlineare thermoviskoelastische Verhalten unter beliebigen Spannungs- und Temperaturverläufen entwickelt und die thermomechanisch gekoppelte Energiegleichung wird angegeben. Im Detail werden der thermodynamisch-einfache Werkstoff, die modifizierte Überlagerung und das isotrope Spannungs-Potenzgesetz diskutiert. Damit thermische Störungen sich mit endlicher Geschwindigkeit ausbreiten, wird ein modifiziertes Fouriersches Wärmeleitgesetz verwendet. Unter Verwendung des nichtlinearen thermoviskoelastischen Spannungs-Potenzgesetzes, der linearisierten Energiegleichung und des modifizierten Wärmeleitgesetzes werden Lösungen der eindimensionalen Wellenfrontausbreitung erhalten.