Supplemental Material: Nonlinear fault damage zone scaling revealed through analog modeling

(1) Movie and description of displacement gradient field evolution; (2) movie of strain fields evolution; (3) cummulative displacement field; (4) displacement gradient profiles through faults; (5) volumetric strain profiles through faults; (6) mechanical properties of the frictional layer; (7) images and parametric data of all the models; (8) a graph of all the measured D-T data; and additional information on the method and scaling to nature.<br>

Supplemental Material: Nonlinear fault damage zone scaling revealed through analog modeling

(1) Movie and description of displacement gradient field evolution; (2) movie of strain fields evolution; (3) cummulative displacement field; (4) displacement gradient profiles through faults; (5) volumetric strain profiles through faults; (6) mechanical properties of the frictional layer; (7) images and parametric data of all the models; (8) a graph of all the measured D-T data; and additional information on the method and scaling to nature.<br>

Balance laws for small deformations

For solids it is often a good approximation to assume that the displacement and displacement gradient are small everywhere in the body. In this setting the basic laws for balance of mass, the balance laws for forces and moments, and the two laws of thermodynamics --- the balance of energy and the imbalance of entropy --- take on simpler forms which are derived and discussed in this chapter.

ENGINEERING MODEL FOR THE CONSOLIDATION OF OIL SEED MATERIAL DURING VISCOPLASTIC DEFORMATION

Cоставлено модельное уравнение консолидации масличного материала, учитывающее процесс его вязкопластичной деформации для неньютоновской твердожидкой смеси. Использованы параллельные соединения моделей Ньютона с вязкостью ηН и Сен-Венана с пределом текучести σ0. Обе формулы представлены в виде конститутивных реологических уравнений состояния. В процессе расчета использовано прямое и обратное преобразования Лапласа. Рассмотрен одномерный вариант пластической деформации масличного материала. Выведены градиенты смещения и скорости. Полученное уравнение распределения давления по высоте слоя материала позволяет определить изменение давления по толщине прессуемого материала с учетом его вязкопластичной деформации. The aim of the study was to compile a model equation for the consolidation of oilseeds, taking into account the process of its viscoplastic deformation for a non-Newtonian solid/liquid mixture. To compile the equation, we used in parallel a combination of Newton models with viscosity ηН and Saint-Venant with yield strength σ0. Both formulas are constitutive rheological equations of state. In the calculation process, the direct and inverse Laplace transforms were used, and a one-dimensional version of the plastic deformation of the oil material was considered, due to which the displacement gradient and velocity gradients were derived. As a result, the obtained equation of pressure distribution over the height of the material layer allows one to determine the pressure change over the thickness of the pressed material, taking into account its viscoplastic deformation.

A COMPLEMENTARY ENERGY THEOREM FOR COMPOSITE PLATES IN POSTBUCKLING

A composite von Karman plate in postbuckling is considered. Using the first Piola stress tensor and the displacement gradient tensor, a complementary energy variational theorem is proven. The proof is given in the case of symmetric lay-up. According to the theorem, at the actual stress state of the plate the complementary energy (as a function of the internal forces and of the moments) reaches its stationary value. The stationary feature of the actual state is valid as compared to other states satisfying the static equilibrium and the static boundary conditions. It is shown how the theorem may be generalized to the case of a non-symmetric lay-up.

A theoretical model for the tread slip and the effective rolling radius of the tyres in free rolling

A rigid–elastic coupling theory for the rolling kinematics of tyres, by which the tread slip, the effective rolling radius and the ply steering of the tyres can be analysed, was formulated. The theory demonstrates that, during free-rolling motion, the coupling between the rotational velocity Ω along the wheel axis and the circumferential displacement gradient generates the longitudinal tread slip, and the coupling between the rotational velocity Ω along the wheel axis and the lateral displacement gradient generates the lateral tread slip and the ply steering. Revisiting the tread slip phenomenon using the newly proposed method numerically proves the kinematic mechanism of the rolling friction. It can be found that the effective rolling radius Re increases with increasing inflation pressure and decreases with increasing load; the sensitivity of Re to the inflation pressure is greater than the sensitivity of Re to the load. The good agreement between the theoretical predictions and the test results for the effective rolling radius of the radial tyres of a car is found. The coupling of the displacement gradient and the rigid rolling kinematics causes non-symmetry of the lateral tread slip which, in turn, leads to the ply steering and the residual aligning moment; these depend on the belt and tread design. The proposed theory and approach provide a quick and powerful tool for analysing the tread slip, the effective rolling radius and the ply steering of the tyres under free rolling.