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Deformation and Strain Stress and strain Strain Lecture 3 – deformation, strain Mechanical Engineering Design - N.Bonora 2018 Stress and strain Introduction • Real bodies are deformable • Under the action of an external loading, the body transforms from a reference configuration to a current configuration. • A “configuration” is a set containing the positions of all particles of the body • This transformation is called: deformation. A deformation may be caused by: • external loads, • body forces (such as gravity or electromagnetic Deformation of a thin rope forces), • changes in temperature, moisture content, or chemical reactions, etc. Mechanical Engineering Design - N.Bonora 2018 Stress and strain Deformation vs strain • Strain is a measure of deformation representing the displacement between particles in the body relative to a reference length. L0 • In a continuous body, a deformation field results Lf from a stress field induced by applied forces or is due to changes in the temperature field inside L0 the body. • The relation between stresses and induced strains is expressed by constitutive equations, e.g., Hooke's law for linear elastic materials. Deformation of a thin rope Mechanical Engineering Design - N.Bonora 2018 Stress and strain Deformation and strain • Strain is a measure of deformation representing the displacement between particles in the body relative to a reference length. L0 • A general deformation of a body can be Lf expressed in the form x=F(X), where X is the reference position of the body L0 • This definition does not distinguish between rigid body motions (translations and rotations) and changes in shape (and size) of the body 휕 Deformation of a thin rope 휺 = 풙 − 푿 = 푭′ − 푰 휕푿 • Strains are dimensionless Mechanical Engineering Design - N.Bonora 2018 Stress and strain Strain measures • Depending on the amount of strain, three main theories are for the analysis of deformation: L0 • Infinitesimal strain theory Lf L • Finite strain theory 0 • Large-displacement (large rotation) theory Deformation of a thin rope Mechanical Engineering Design - N.Bonora 2018 Stress and strain Strain measures • Infinitesimal strain theory • Also called small strain theory, small deformation theory, small displacement L theory, or small displacement-gradient 0 theory. 퐿푓 = 퐿표 + Δ퐿 • Strains and rotations are both small. The undeformed and deformed configurations of the body can be assumed identical. • The infinitesimal strain theory is used in the analysis of deformations of materials exhibiting elastic behavior, such as materials found in mechanical and civil engineering applications, e.g. concrete and steel. Mechanical Engineering Design - N.Bonora 2018 Stress and strain Strain measures • Finite strain theory • also called large strain theory, large deformation theory L0 • Deals with deformations in which both rotations and strains are arbitrarily large. In this case, the undeformed and deformed configurations of the continuum are 퐿푓 significantly different and a clear distinction has to be made between them. • This is commonly the case with elastomers, plastically-deforming materials and other fluids and biological soft tissue. Mechanical Engineering Design - N.Bonora 2018 Stress and strain Strain measures • Large strain theory • Large-displacement or large-rotation theory, which assumes small strains but large rotations and displacements. L0 퐿푓 Mechanical Engineering Design - N.Bonora 2018 Stress and strain Strain measures • In each theory the definition of strain is different 퐿0 • Engineering strain. The Cauchy strain or engineering strain is expressed as the ratio of 퐿푓 total deformation to the initial dimension of the material body in which the forces are being applied. • Engineering normal strain of a material line 퐿푓 − 퐿0 휀 = element or fiber axially loaded is given by 퐿0 the change in length ΔL per unit of the original length L of the line element or fibers. The normal strain is positive if the 휕푢 material fibers are stretched and negative if 퐿 ≈ 푑푥 + 푑푥 푓 휕푥 they are compressed. Mechanical Engineering Design - N.Bonora 2018 Stress and strain Strain measures • In each theory the definition of strain is different • Engineering strain. The Cauchy strain or engineering strain is expressed as the ratio of total deformation to the initial dimension of the material body in which the forces are being applied. • Engineering shear strain is defined as the change in angle between lines AC and AB. 휕푢푦 휕푢 훾 = + 푥 푥푦 휕푥 휕푦 Mechanical Engineering Design - N.Bonora 2018 Stress and strain Strain measures • Different definitions Δ푙 • Engineering strain: 휀푒푛푔 = 푙0 푑푙 Δ푙 • True engineering strain 휀 = = 푙푛 1 + 푙 푙0 2 2 • Green strain 1 푙 − 푙0 휀퐺 = 2 2 푙0 2 2 1 푙 − 푙0 • Almansi strain 휀퐺 = 2 푙2 Mechanical Engineering Design - N.Bonora 2018 Stress and strain Examples of homogeneous deformation Straight lines remain straight, parallel lines remains parallel • uniform extension 휀 0 휀 = • pure dilation 0 휀 • simple shear • pure shear Mechanical Engineering Design - N.Bonora 2018 Stress and strain Strain tensor STRAIN TENSOR (infinitesimal deformations). A tridimensional deformation state is described by the strain tensor e 휀11 휀12 휀13 휀푖푗 = 휀21 휀22 휀23 휀31 휀32 휀33 Same properties as for the stress tensor: strain invariants 퐼1 = 휀11 + 휀22 + 휀33 2 2 2 퐼2 = 휀11휀22 + 휀22휀33 + 휀33휀11 − 휀12 − 휀23 − 휀31 퐼3 = 푑푒푡 휀푖푗 Mechanical Engineering Design - N.Bonora 2018 Stress and strain Strain tensor PRINCIPAL STRAIN. At every point in a deformed VOLUMETRIC STRAIN. The dilatation (the body there are at least three planes, called relative variation of the volume) is the trace of principal planes, with normal vectors, called the tensor: principal directions, where the corresponding strain vector is perpendicular to the plane and Δ푉 휀푘푘 = 훿 = = 휀11+휀22+휀33 where there are no shear strain. 푉0 The three strain normal to these principal planes are called principal strains. 휀11 0 0 휀푖푗 = 0 휀22 0 0 0 휀33 Mechanical Engineering Design - N.Bonora 2018 Stress and strain Strain tensor PRINCIPAL STRAIN. At every point in a deformed STRAIN DEVIATOR. The infinitesimal strain body there are at least three planes, called tensor, similarly to the Cauchy stress tensor, principal planes, with normal vectors, called can be expressed as the sum of two other principal directions, where the corresponding tensors. The strain deviator account for strain vector is perpendicular to the plane and distortion. where there are no shear strain. 1 휀 = 휀′ + 휀 훿 The three strain normal to these principal 푖푗 푖푗 3 푘푘 푖푗 planes are called principal strains. 휀 − 훿 휀 휀 휀11 0 0 11 12 13 휀′ =≡ 휀21 휀푦 − 훿 휀23 휀푖푗 = 0 휀22 0 푖푗 0 0 휀33 휀31 휀32 휀푧 − 훿 Mechanical Engineering Design - N.Bonora 2018 Stress and strain Strain tensor OCTAHEDRAL STRAINS. An octahedral plane is one whose normal makes equal angles with the three principal directions. The engineering shear strain on an octahedral plane is called the octahedral shear strain and is given by 2 훾 = 휀 − 휀 2 + 휀 − 휀 2 + 휀 − 휀 2 표푐푡 3 1 2 2 3 3 1 Mechanical Engineering Design - N.Bonora 2018 Stress and strain Strain tensor: special cases PLANE STRAIN. The plane strain condition is ANTIPLANE STRAIN. This state of strain is when there are no shear strain in any plane achieved when the displacements in the body normal to the plane considered and the strain are zero in the plane of interest but nonzero in component normal to such plane is zero the direction perpendicular to the plane. 휀11 휀12 0 0 0 휀13 휀 = 휀 휀 0 푖푗 21 22 휀푖푗 = 0 0 휀23 0 0 0 휀31 휀32 0 Mechanical Engineering Design - N.Bonora 2018 Stress and strain Symmetry and additivity Mechanical Engineering Design - N.Bonora 2018 Stress and strain Symmetry and additivity Mechanical Engineering Design - N.Bonora 2018 Stress and strain STRAIN RATE 푙푓 − 푙0 푑휀 푑 1 푑푙 푣 Engineering strain rate 푒푛푔 푙0 휀푒푛푔 = = = = 푑푡 푑푡 푙0 푑푡 푙0 푙 푑 푙푛 푙0 1 푑푙 푣 True strain rate 휀 == = = 푑푡 푙 푑푡 푙 Mechanical Engineering Design - N.Bonora 2018 Stress and strain Suggested readings • http://www.mech.utah.edu/~brannon/public/Mohrs_Circle.pdf • Schaum's Outline of Strength of Materials, Fifth Edition (Schaum's Outline Series) Fifth (5th) Edition Paperback – September 12, 2010 • Strength of Materials (Dover Books on Physics) Reprinted Edition by J. P. Den Hartog, ISBN-10: 0486607550 Mechanical Engineering Design - N.Bonora 2018.
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