Nonuniform Stress Field Determination Based on Deformation Measurement

2021 ◽  
pp. 1-26
Author(s):  
Cheng Liu
Keyword(s):  
Principal Stress ◽  
Deformation Gradient ◽  
Deformation Measurement ◽  
Cauchy Stress ◽  
Kirchhoff Stress ◽  
Nonuniform Deformation ◽  
Principal Stretch ◽  
Stress Orientation ◽  
Lagrangian Strain ◽  
Stretch Orientation

Abstract We demonstrate a technique that, under certain circumstances, will determine stresses associated with a nonuniform deformation field without knowing the detailed constitutive behavior of the deforming material. This technique is based on (1) a detailed deformation measurement of a domain and (2) the observation that for isotropic materials, the strain and the stress, which form the so-called work-conjugate pair, are co-axial, or their eigenvectors share the same direction. The particular measures for strain and stress considered are the Lagrangian strain and the second Piola-Kirchhoff stress. The deformation measurement provides the field of the principal stretch orientation θλ and since the Lagrangian strain and the second Piola-Kirchhoff stress are co-axial, the principal stress orientation θs of the second Piola-Kirchhoff stress is determined. The Cauchy stress is related to the second Piola-Kirchhoff stress through the deformation gradient tensor, which can be measured experimentally. We then show that the principal stress orientation θσ of the Cauchy stress is the sum of the principal stretch orientation θλ and the local rigid-body rotation θq, which is determinable by the deformation gradient through polar decomposition. With the principal stress orientation θσ known, the equation of equilibrium, now in terms of the two principal stresses σ1 and σ2, and θσ, can be solved numerically with appropriate traction boundary conditions. The technique is then applied to the experimental case of nonuniform deformation of a PVC sheet with a circular hole and subject to tension. Limitations and restrictions of the technique and possible extensions will be discussed.

Revisit on the analytic formulations of principal stress orientation in fracture mechanics

2020 ◽  
Author(s):  
Arjun Ajit Kottara ◽  
Govind Padmanabhan ◽  
M. Maneesh Kumar ◽  
K. Rohit ◽  
M.P. Hariprasad
Keyword(s):  
Fracture Mechanics ◽  
Principal Stress ◽  
Stress Orientation

The effect of principal stress orientation on tunnel stability

2015 ◽  
Vol 49 ◽  
pp. 279-286 ◽  
Author(s):  
Zheming Zhu ◽  
Yuanxin Li ◽  
Jun Xie ◽  
Bang Liu
Keyword(s):  
Principal Stress ◽  
Tunnel Stability ◽  
Stress Orientation

scholarly journals A novel thermo-mechanical coupling approach for thermal fracturing of rocks in the three-dimensional FDEM

2020 ◽  
Vol 7 (5) ◽  
pp. 935-946 ◽  
Author(s):  
Clément Joulin ◽  
Jiansheng Xiang ◽  
John-Paul Latham
Keyword(s):  
Thermal Expansion ◽  
Deformation Gradient ◽  
Linear Thermal Expansion ◽  
Three Dimensional ◽  
Cauchy Stress ◽  
Reinforced Concrete Structure ◽  
Expansion Formula ◽  
Mechanical Coupling ◽  
Coupling Approach ◽  
Thermal Fracturing

Abstract This paper presents a new three-dimensional thermo-mechanical (TM) coupling approach for thermal fracturing of rocks in the finite–discrete element method (FDEM). The linear thermal expansion formula is implemented in the context of FDEM according to the concept of the multiplicative split of the deformation gradient. The presented TM formulation is derived in the geo-mechanical solver, enabling thermal expansion and thermally induced fracturing. This TM approach is validated against analytical solutions of the Cauchy stress, thermal expansion and stress distribution. Additionally, the thermal load on the previously validated configurations is increased and the resulting fracture initiation and propagation are observed. Finally, simulation results of the cracking of a reinforced concrete structure under thermal stress are compared to experimental results. Results are in excellent agreement.

scholarly journals Understanding the deformation gradient in Abaqus and key guidelines for anisotropic hyperelastic user material subroutines (UMATs)

2019 ◽  
Author(s):  
David Nolan ◽  
Caitriona Lally ◽  
Patrick McGarry
Keyword(s):  
Local Coordinate System ◽  
Deformation Gradient ◽  
Worked Examples ◽  
Local Deformation ◽  
Cauchy Stress ◽  
Precise Understanding ◽  
Key Issues ◽  
Key Steps ◽  
Orientation Systems ◽  
Global And Local

This tutorial paper provides a step-by-step guide to developing a comprehensive understanding of the different forms of the deformation gradient used in Abaqus, and outlines a number of key issues that must be considered when developing an Abaqus user defined material subroutine (UMAT) in which the Cauchy stress is computed from the deformation gradient. Firstly, we examine the "classical" forms of global and local deformation gradients. We then show that Abaqus/Standard does not use the classical form of the local deformation gradient when continuum elements are used, and we highlight the important implications for UMAT development. We outline the key steps that must be implemented in developing an anisotropic fibre-reinforced hyperelastic UMAT for use with continuum elements and local orientation systems. We also demonstrate that a classical local deformation gradient is provided by Abaqus/Standard if structural (shell and membrane) elements are used, and by Abaqus/Explicit for all element types. We emphasise, however, that the majority of biomechanical simulations rely on the use of continuum elements with a local coordinate system in Abaqus/Standard, and therefore the development of a hyperelastic UMAT requires an in-depth and precise understanding of the form of the non-classical deformation gradient provided as input by Abaqus. Several worked examples and case studies are provided for each section, so that the details and implications of the form of the deformation gradient can be fully understood. For each worked example in this tutorial paper the source files and code (Abaqus input files, UMATs, and Matlab script files) are provided, allowing the reader to efficiently explore the implications of the form of the deformation gradient in the development of a UMAT.

scholarly journals Initial stress symmetry and its applications in elasticity

2015 ◽  
Vol 471 (2183) ◽  
pp. 20150448 ◽  
Author(s):  
A. L. Gower ◽  
P. Ciarletta ◽  
M. Destrade
Keyword(s):  
Residual Stress ◽  
Free Energy ◽  
Thermal Expansion ◽  
Initial Stress ◽  
Elastic Deformation ◽  
Deformation Gradient ◽  
Free Energy Density ◽  
Cauchy Stress ◽  
Optimal Target ◽  
External Loads

An initial stress within a solid can arise to support external loads or from processes such as thermal expansion in inert matter or growth and remodelling in living materials. For this reason, it is useful to develop a mechanical framework of initially stressed solids irrespective of how this stress formed. An ideal way to do this is to write the free energy density Ψ in terms of initial stress τ and the elastic deformation gradient F , so we write Ψ = Ψ ( F , τ ). In this paper, we present a new constitutive condition for initially stressed materials, which we call the initial stress symmetry (ISS). We focus on two consequences of this condition. First, we examine how ISS restricts the possible choices of free energy densities Ψ = Ψ ( F , τ ) and present two examples of Ψ that satisfy the ISS. Second, we show that the initial stress can be derived from the Cauchy stress and the elastic deformation gradient. To illustrate, we take an example from biomechanics and calculate the optimal Cauchy stress within an artery subjected to internal pressure. We then use ISS to derive the optimal target residual stress for the material to achieve after remodelling, which links nicely with the notion of homeostasis.

scholarly journals Design Arbitrary Shaped 2D Acoustic Cloak Without Singularity

2009 ◽  
Author(s):  
Jin Hu ◽  
Xiaoming Zhou ◽  
Gengkai Hu
Keyword(s):  
Deformation Gradient ◽  
Transformation Method ◽  
Two Dimensional ◽  
Gradient Tensor ◽  
Deformation Gradient Tensor ◽  
Material Parameters ◽  
Principal Stretch ◽  
Irregular Shapes ◽  
Principal Stretches ◽  
Analytical Expressions

A method is proposed to design arbitrary shaped two dimensional (2D) isotropic-inertia acoustic cloaks without singularity. The method is based on the deformation view of the transformation method, where the transformation tensor A is identified as the deformation gradient tensor and the transformed material parameters can be expressed by the principal stretches in the principal system of the deformation. The infinite material parameters of a perfect 2D cloak is induced by an infinite principal stretch in one direction while the other two remains finite at the inner boundary during the transformation. To circumvent this difficulty, for a 2D cloak we can choose the principal stretch perpendicular to the cloak plane to be also infinite but in the same order as the infinite principal stretch in the cloak plane during the transformation, so the transformed material parameters may keep finite. To illustrate this idea, the analytical expressions of nonsingular material parameters for a cylindrical acoustic cloak are given. For the acoustic cloaks with irregular shapes, the numerical method is proposed to evaluate the principal stretches and in turn the nonsingular material parameters. The designed 2D cloaks are validated by numerical simulation.

Some influences of stratigraphy and structure on reservoir stress orientation

Geophysics ◽  
1994 ◽  
Vol 59 (6) ◽  
pp. 954-962 ◽  
Author(s):  
Michael S. Bruno ◽  
Don F. Winterstein
Keyword(s):  
Principal Stress ◽  
Compressive Stress ◽  
Horizontal Stress ◽  
Oil Fields ◽  
Fold Axis ◽  
Principal Stresses ◽  
Stress Orientation ◽  
Stress Direction ◽  
Well Pattern ◽  
Maximum Horizontal Stress

The azimuth of maximum horizontal stress in a reservoir can vary significantly with depth and with position on a subsurface structure. We present and discuss evidence from field data for such variation and demonstrate both analytically and with finite‐element modeling how such changes might take place. Under boundary conditions of uniform far‐field displacement, changes in stratigraphic layering can reorient the principal stress direction if the formation is intrinsically anisotropic. If the formation stiffness is lower perpendicular to bedding than parallel to bedding (as is often the case in layered geologic media), an increase in dip will reduce the component of compressive stress in the dip azimuth direction. Folds can reorient principal stresses because flexural strain varies with depth and position. Compressive stress perpendicular to a fold axis increases with depth at the crest of an anticline and decreases with depth at the limb. When the regional stress anisotropy is weak, this change in stress magnitude can reorient the local principal stress directions. Numerical simulations of such effects gave results consistent with changes in stress orientation at the Cymric and Lost Hills oil fields in California as observed via shear‐wave polarization analyses and tiltmeter surveys of hydraulic fracturing. Knowledge of such variation of stress direction with depth and structural position is critical for drilling, completions, hydraulic fracture, and well pattern designs.

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