Inter-Element Crack Propagation with High-Order Stress Equilibrium Element

The present contribution proposes a formulation based on the use of hybrid equilibrium elements (HEEs), for the analysis of inter-element delamination and fracture propagation problems. HEEs are defined in terms of quadratic stress fields, which strongly verify both the homogeneous and inter-element equilibrium equations and they are employed with interfaces, initially exhibiting rigid behavior, embedded at the elements’ sides. The interface model is formulated in terms of the same degrees of freedom of the HEE, without any additional burden. The cohesive zone model (CZM) of the extrinsic interface is rigorously developed in the damage mechanics framework, with perfect adhesion at the pre-failure condition and with linear softening at the post-failure regime. After a brief review, the formulation is computationally tested by simulating the behavior of a double-cantilever-beam with diagonal loads; the obtained numerical results confirm the accuracy and potential of the method.

Geometry Effects on Mode I Brittle Fracture in VO-Notched PMMA Specimens

This paper gathers experimental and theoretical investigations about both the geometry-dependent fracture initiation angle and the fracture strength in VO-notched polymethyl methacrylate (PMMA) specimens under mode I loading conditions. The numerical analyses revealed that despite the application of pure mode I loading on the geometrically symmetric VO-notched samples, the maximum tangential stress occurs at two points symmetrically placed on either side of the notch bisector line. The experimental tests performed on some specimens showed that a crack does not necessarily propagate along the notch bisector line. Stress-based theoretical studies were then carried out to justify the experimental findings. The conventional maximum tangential stress (MTS) criterion gave weak predictions of the fracture. Therefore, the predictions were checked with the generalized MTS (GMTS) criterion by taking into consideration the higher-order stress terms. It was demonstrated that the GMTS criterion predictions have satisfactory consistency with the experimental results of the crack initiation angle and the fracture strength.

Stress rotation – impact and interaction of rock stiffness and faults

It has been assumed that the orientation of the maximum horizontal compressive stress (SHmax) in the upper crust is governed on a regional scale by the same forces that drive plate motion. However, several regions are identified where stress orientation deviates from the expected orientation due to plate boundary forces (first-order stress sources), or the plate wide pattern. In some of these regions, a gradual rotation of the SHmax orientation has been observed. Several second- and third-order stress sources have been identified in the past, which may explain stress rotation in the upper crust. For example, lateral heterogeneities in the crust, such as density and petrophysical properties, and discontinuities, such as faults, are identified as potential candidates to cause lateral stress rotations. To investigate several of these candidates, generic geomechanical numerical models are set up with up to five different units, oriented by an angle of 60∘ to the direction of shortening. These units have variable (elastic) material properties, such as Young's modulus, Poisson's ratio and density. In addition, the units can be separated by contact surfaces that allow them to slide along these vertical faults, depending on a chosen coefficient of friction. The model results indicate that a density contrast or the variation of Poisson's ratio alone hardly rotates the horizontal stress (≦17∘). Conversely, a contrast of Young's modulus allows significant stress rotations of up to 78∘, even beyond the vicinity of the material transition (>10 km). Stress rotation clearly decreases for the same stiffness contrast, when the units are separated by low-friction discontinuities (only 19∘ in contrast to 78∘). Low-friction discontinuities in homogeneous models do not change the stress pattern at all away from the fault (>10 km); the stress pattern is nearly identical to a model without any active faults. This indicates that material contrasts are capable of producing significant stress rotation for larger areas in the crust. Active faults that separate such material contrasts have the opposite effect – they tend to compensate for stress rotations.

Lithospheric-scale anisotropies control first-order stress orientation during Cretaceous-Cenozoic plate kinematics in Western-Central Europe

<p>Late Mesozoic-Cenozoic plate convergence led to widespread intraplate deformation in Western-Central Europe during the Late Cretaceous-Paleogene and the Miocene until today reflecting the collision of Eurasia with Iberia-Africa and Adria, respectively. The resulting complex deformation pattern inside the plate boundary zone contrasts with a rather uniform orientation adjacent to the north. Although there is broad consensus that the orientation of the first-order stress is controlled by plate kinematics, there is no sufficient explanation for the variation of the stress field across the plate boundary. We model plate kinematic trajectories and analyze the spatial distribution of paleostress data from fault-slip inversion and tectonic stylolites. The comparison reveals the coexistence of two contrasting stress provinces in Europe throughout the Late Mesozoic-Cenozoic. Inside the diffuse plate boundary zone, trajectories of plate motion fit deformation patterns. Outside of that zone, however, there is significant deviation. Here deformation is mainly accommodated by the reactivation of Paleozoic shear zones. Thus, we argue that lithospheric-scale structural inheritance from the Pangea assemblage controls the stress-strain pattern of Western-Central Europe between the active plate boundary zone and the East European Craton since the Late Mesozoic.</p>

Phase Field Modeling of Brittle Fracture Based on the Cell-Based Smooth FEM by Considering Spectral Decomposition

The spectral decomposition of the strain tensor is an essential technique to deal with the fracture problems via phase field method, and some incorrect results may be obtained without it. A novel phase field model for brittle fracture is developed based on cell-based smooth finite element (CS-FEM) and the spectral decomposition is taken into account. In order to describe the nonlinearity behaviors which contain the varied stress and elastic constitutive response caused by spectral decomposition. A second-order stress tensor and a fourth-order constitutive tensor based on decomposition of strain tensor are derived. A fundamental framework of CS-FEM is established to solve the phase field fracture problems, implemented by user-defined element (UEL) subroutine of ABAQUS software. The proposed model is validated by a typical Mode II crack, and the results show that the derived tensors are effective. Phase field parameter, CS-FEM parameter and mesh inhomogeneity are investigated to provide some useful suggestion for further development. Some classical numerical examples are solved by using the present model. The studies demonstrate that the proposed method can successfully overcome mesh distortion; the number of smooth cell does not show influences on the accuracy. Moreover, some results show that this method has the advantage over the standard FEM in convergence and computing efficiency.

Stress rotation – The impact and interaction of rock stiffness and faults

It has been assumed, that the maximum compressive horizontal stress (SHmax) orientation in the upper crust is governed on a regional scale by the same forces that drive plate motion. However, several regions are identified, where stress orientation deviates from the expected orientation due to plate boundary forces (first order stress sources), or the plate wide pattern. In some of this regions a gradual rotation of the SHmax orientation has been observed. Several second and third order stress sources have been identified, which may explain stress rotation in the upper crust. For example lateral heterogeneities in the crust, such as density, petrophysical or petrothermal properties and discontinuities, like faults are identified as potential candidates to cause lateral stress rotations. To investigate several of the candidates, generic geomechanical numerical models are utilized. These models consist of up to five different units, oriented by an angle of 60° to the direction of contraction. These units have variable elastic material properties, such as Young's modulus, Poisson ratio and density. Furthermore, the units can be separated by contact surfaces that allow them so slide along these faults, depending on a selected coefficient of friction. The model results indicate, that a density contrast or the variation of the Poisson's ratio alone sparsely rotates the horizontal stress orientation (≦ 17°). Conversely, a contrast of the Young's modulus allows significant stress rotations in the order of up to 78°; not only areas in the vicinity of the material transition are affected by that stress rotation. Stress rotation clearly decreases for the same stiffness contrast, when the units are separated by low friction discontinuities (19°). Low friction discontinuities in homogeneous models do not change the stress pattern at all, away from the fault; the stress pattern is nearly identical to a model without any active faults. This indicates that material contrasts are capable of producing significant stress rotation for larger areas in the crust. Active faults that separates such material contrasts have the opposite effect, they rather compensate stress rotations.