DIFFERENTIAL EQUATIONS OF EQUILIBRIUM OF IDEALLY ELASTOPLASTIC CONTINUOUS MEDIUM FOR PLANE ONE-DIMENSIONAL DEFORMATION IN APPROXIMATION OF CLOSING EQUATIONS BY BIQUADRATIC FUNCTIONS

The present article considers the construction of differential equations of equilibrium of geometrically and physically nonlinear ideally elastoplastic in relation to shear deformations of continuous medium under conditions of one-dimensional plane deformation, when the diagrams of volumetric and shear deformation are approximated by biquadratic functions. The construction of physical dependencies is based on calculating the secant moduli of volumetric and shear deformation. When approximating the graphs of the volumetric and shear deformation diagrams using two segments of parabolas, the secant shear modulus in the first segment is a linear function of the intensity of shear deformations; the secant modulus of volumetric expansion-contraction is a linear function of the first invariant of the strain tensor. In the second section of the diagrams of both volumetric and shear deformation, the secant shear modulus is a fractional (rational) function of the shear strain intensity; the secant modulus of volumetric expansion-compression is a fractional (rational) function of the first invariant of the strain tensor. Based on the assumption of independence, generally speaking, from each other of the volumetric and shear deformation diagrams, five main cases of physical dependences are considered, depending on the relative position of the break points of the graphs of the diagrams volumetric and shear deformation. On the basis of received physical equations, differential equations of equilibrium in displacements for continuous medium are derived under conditions of plane one-dimensional deformation. Differential equations of equilibrium in displacements constructed in the present article can be applied in determining stress and strain state of geometrically and physically nonlinear ideally elastoplastic in relation to shear deformations of continuous medium under conditions of plane one-dimensional deformation, closing equations of physical relations for which, based on experimental data, are approximated by biquadratic functions.

Differential equations of equilibrium of continuous medium for plane one-dimensional deformation at closing equations approximation by biquadratic functions

Problems of differential equations construction of equilibrium of a geometrically and physically nonlinear continuous medium under conditions of one-dimensional plane deformation are considered, when the diagrams of volumetric and shear deformation are approximated by quadratic functions. The construction of physical dependencies is based on calculating the secant moduli of volumetric and shear deformation. When approximating the graphs of the volumetric and shear deformation diagrams using two segments of parabolas, the secant shear modulus in the first segment is a linear function of the intensity of shear deformations, the secant modulus of volumetric expansion - contraction is a linear function of the first invariant of the strain tensor. In the second section of the diagrams of both volumetric and shear deformation, the secant shear modulus is a fractional (rational) function of the shear strain intensity, the secant modulus of volumetric expansion - compression is a fractional (rational) function of the first invariant of the strain tensor. Based on the assumption of independence, generally speaking, from each other of the volumetric and shear deformation diagrams, six main cases of physical dependences are considered, depending on the relative position of the break points of the graphs of the diagrams volumetric and shear deformation, each approximated by two parabolas. The differential equations of equilibrium in displacements constructed in the article can be applied in determining the stressed and deformed state of a continuous medium under conditions of one-dimensional plane deformation, the closing equations of physical relations for which, constructed on the basis of experimental data, are approximated by biquadratic functions.

Effects of gravel content on liquefaction resistance and its assessment considering deformation characteristics in gravel – mixed sand

Many reports describe overestimation of liquefaction resistance based on sounding data related to ground materials containing coarse particles such as gravel and cobbles. Better methods of liquefaction potential estimation must be developed using investigation data other than those from sounding. Gathering perfect and undisturbed samples is difficult, but using seismic methods such as PS logging might be effective for assessing liquefaction potential. For this study, bender element (BE) tests and local small strain (LSS) tests were conducted, respectively, to measure the dynamic and static shear moduli of gravel – mixed sand specimens. Subsequently, relations between liquefaction strength and secant shear moduli were examined to provide reliable estimation of liquefaction in gravel – mixed sand. Although the liquefaction resistance increased considerably with overconsolidation, the initial shear modulus exhibited only a slight change with the same overconsolidation. The experimentally obtained results elucidated that the important shear strain level, for which secant shear modulus has a strong relation with liquefaction strength, was not a linear elastic region of 0.001%: it was about 0.01%.