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1: The Concept of Strain

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    Strain is a fundamental concept in continuum and structural mechanics. Displacement fields and strains can be directly measured using gauge clips or the Digital Image Correlation (DIC) method. Deformation patterns for solids and deflection shapes of structures can be easily visualized and are also predictable with some experience. By contrast, the stresses can only be determined indirectly from the measured forces or by the inverse engineering method through a detailed numerical simulation. Furthermore, a precise determination of strain serves to define a corresponding stress through the work conjugacy principle. Finally the equilibrium equation can be derived by considering compatible fields of strain and displacement increments, as explained in Chapter 2. The present author sees the engineering world through the magnitude and shape of the deforming bodies. This point of view will dominate the formulation and derivation throughout the present lecture note. Chapter 1 starts with the definition of one dimensional strain. Then the concept of the threedimensional (3-D) strain tensor is introduced and several limiting cases are discussed. This is followed by the analysis of strains-displacement relations in beams (1-D) and plates (2- D). The case of the so-called moderately large deflection calls for considering the geometric non-linearities arising from rotation of structural elements. Finally, the components of the strain tensor will be re-defined in the polar and cylindrical coordinate system.

    • 1.1: One-dimensional Strain
      This page explores the behavior of a fixed prismatic rod under tensile forces, emphasizing homogeneous strain in uniform deformation scenarios. It presents different strain definitions—engineering, Cauchy, and logarithmic—highlighting their equivalence under small displacements. Additionally, it underscores the importance of local strain definitions, illustrating the differences in strain within non-homogeneous fields through displacement gradients.
    • 1.2: Extension to the 3-D case
      This page covers displacement gradients, strain tensors, and spin tensors in three-dimensional space. It defines the displacement gradient as a non-symmetric tensor, encompassing rigid body rotation, while distinguishing the strain tensor as its symmetric part, eliminating rotation’s impact.
    • 1.3: Description of Strain in the Cylindrical Coordinate System
      This page derives strain-displacement relations in cylindrical coordinates, highlighting normal and shear components of the strain tensor. It covers calculations for radial and circumferential strains, along with shear strains in different planes, concluding with the six components of the infinitesimal strain tensor. The page also simplifies equations for axial symmetry, with future chapters planned to explore applications to circular plates and cylindrical shells.
    • 1.4: Kinematics of the Elementary Beam Theory
      This page introduces kinematics in structural mechanics, covering the motion of bodies and key concepts like displacement, strain, and velocity. It differentiates between kinematic and static analysis, noting that static analysis focuses on stresses without motion. The section defines beams, highlighting slenderness ratios and categorizing them into Euler and Timoshenko types based on geometry.
    • 1.5: Euler-Bernoulli Hypothesis
      This page covers fundamental concepts in beam theory, emphasizing hypotheses critical for understanding beam deformation, such as the "plane remains plane" and "normal remains normal" hypotheses. It introduces three key hypotheses that lead to the Euler-Bernoulli theory for analyzing bending strains in beams. The text details how to calculate strain components, noting that only axial strain is considered, and it defines curvature related to beam deformation.
    • 1.6: Strain-Displacement Relation of Thin Plates
      This page outlines the prerequisite relationship between course 2.080 and advanced course 2.081 on Plates and Shells, emphasizing accessible lecture notes through OpenCourseWare. It introduces tensorial and expanded notations for problem-solving, covers representations of points and displacement vectors, and contrasts plate theory with beam theory, noting that plate theory entails fewer assumptions and is simplified by its two-dimensional framework.
    • 1.7: Advanced Topic- Derivation of the Strain-Displacement Relation for Thin Plates
      This page discusses the Love-Kirchoff hypothesis, which expands the Euler-Bernoulli theory to include plates with double curvature. It highlights how the displacement vector's in-plane components vary with the vertical coordinate and analyzes strain components, noting the absence of out-of-plane shear strains. The strain-displacement relation incorporates both membrane action and curvature, linking strain to in-plane displacements and curvature through a defined curvature tensor.
    • 1.8: Expanded Form of Strain-Displacement Relation
      This page explores tensor geometric relationships and their physical interpretations, particularly in relation to strain and curvature within plate theory. It details membrane strains resulting from displacements, differentiating small and moderately large deflections.
    • 1.9: Moderately Large Deflections of Beams and Plates
      This page discusses the theory of moderately large deflections in plates, contrasting it with small deflection scenarios. It categorizes three cases based on beam geometry and summarizes the strain-displacement relations, emphasizing the effects of rotation and extension on strain calculations. The page notes the complexity of nonlinear strain terms involving interactions between deflection slopes. It highlights the significance of these concepts for practical engineering applications.
    • 1.10: Strain-Displacement Relations for Circulate Plates
      This page covers the theory of circular plates in cylindrical coordinates, emphasizing axi-symmetric deformation like zero circumferential displacement and the lack of in-plane shear strains. It identifies radial and circumferential strains as principal strains, with governing equations reducing to ordinary differential equations for closed-form solutions under different boundary conditions.

    This page titled 1: The Concept of Strain is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Tomasz Wierzbicki (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform.