Stress, strain and deformation in solids free download as word doc. Mechanisms of deformation in triangular honeycombs parallel those in 3d trusses lattice materials stress strain curves and deformation behavior. Understand the mathematical and physical foundations of the continuum mechanics of solids, including deformation and stress measures, elastic and plastic stress strain relations, and failure criteria. When an object has a large value of elastic modulus, the effect of stress is small. Mechanics of solids finite deformation and strain tensors. But, if the deformation of the body is prevented, some stresses are. Strain deformation strain is defined as deformation of a solid due to stress. A strain gauge is made by sealing a length of very fine wire in a small rectangle of thin plastic sheet in such a way that if the plastic is stretched put under strain, then the wire will also be stretched. Mechanics of solid stress and strain by kaushal patel 2. Stress is the quantity that is proportional to the external force causing the deformation. The magnitude of the resisting force is numerically equal to the applied force.
The ratio of stress and strain is constant with in the elastic limit. Once the load is increased further, the stress starting exceeding the yield strength. Able to sustain plastic deformation high strain but low stress gradual stages prior to rupture safe steels, plastic brittle the opposite of ductile no gradual stages before rupture concrete, alloys, bones, cold steel strain hardening or strain stiffening strain softening ductile material. Stress strain relations for linearly elastic solids, generalized hookes law. In the theory of finite deformations, extension and rotations of line elements are unrestricted as to size.
In our derivations that follow, we limit our attention to two dimensions. Therefore, there are 6 independent variables in the strain matrix, instead of 9. Fhwa nhi06088 2 stress and strain in soils soils and foundations volume i 2 1 december 2006 chapter 2. For an infinitesimal fibre that deforms from an initial point given by the vector dx to the vector dx in the time t, the deformation gradient is defined by fij. Viscoelastic material behavior is compared to elastic solids and viscous fluids.
Basic concepts of stress and strain mechanics of materials describes how solid materials will deform. The study of temporary or elastic deformation in the case of engineering strain is applied to materials used in mechanical and structural engineering, such as concrete and steel, which are subjected to very small deformations. A plastic deformation occurs when strain is not proportional to stress but when the load is removed the material returns to its original length. This video is ed by the jeff hanson for the private use of our audience. Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration. Explain the concepts of stress and strain in describing elastic deformations of materials describe. An exception to this was the varying stress field in the loaded beam, but there a simplified set of elasticity equations was used. The relationship of stress and strain for steel can be observed in.
Stress is the external force acting on the object per unit area. Couple stress based strain gradient theory for elasticity. Stressstrain curves are an extremely important graphical measure of a materials mechanical. This means that the strain increases rapidly even for a small change in the stress. For sufficiently small stresses, the stress is directly.
This reversible behavior often shows a linear relation between stress and strain. Six ideal model 1 rigid material i a rigid material is one which has no strain regardless of the applied stress. D stress strain 23 nylon breaks when the stress within it reaches 1 x 10 9 pa. Here authors measure the solid surface stress of a gel, and show its dependence on surface strain through a surface modulus. The deformation of an object is typically a change in length. The resistance in crystals is greater than that in amorphous materials. A strain is the measure of the degree of deformation. Youngs modulus is defined as the ratio of stress to strain within elastic limit. The point b in the curve is the yield point or the elastic limit and the corresponding stress is the yield strength s y of the material. Engineering strain is modeled by infinitesimal strain theory, also. In the absence of stretch and dilatation gradients, the size dependence can be explained using classical couple stress theory in which the full curvature tensor is used as deformation measures in addition to the conventional strain measures. A change in the shape of an object due to the application of force is known as deformation. A deformation may be caused by external loads, body forces such as gravity or electromagnetic forces, or changes in temperature, moisture content, or chemical reactions, etc. Strain is the deformation per unit of the original length.
In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighbouring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material. Deformation of solids deformation engineering young. Generalized hookes law for solids with multiaxial stresses. When the stress is sufficient to permanently deform the metal, it is called plastic deformation. The liquid and gas phases occupy the voids between the solid particles as shown in figure 21a. Definition of stress, stress tensor, normal and shear stresses in axially loaded members. The deformation behavior of materials in the micron scale has been experimentally shown to be size dependent. Stress and strain mechanics of solid linkedin slideshare. Lectures notes on mechanics of solids course code bme203 prepared by prof. Nonlinear stress strain relations can be approximated by including third and higher order terms of the strain tensor in the elastic energy expression e. Thus, a rigid body displacement induces no strain, and hence no stress, in the solid. Stress, strain and youngs modulus engineering toolbox. Students learn about materials that have viscoelastic behavior along with the importance of engineers.
Stress when an external force acts on a body, it undergoes deformation. Some, such as plant structures and coastlines, may be so arbitrary as to defy. Strain displacement relations, equilibrium equations, compatibility conditions and airys stress function. Ppt stress and strain powerpoint presentation free to. Show that the lagrange strain associated with this deformation is. Modulus of elasticity the modulus of elasticity youngs modulus e is a material property, that describes its stiffness and is therefore one of the most important properties of solid materials. Stress is force per unit area strain is the deformation of a solid due to stress.
Stress and strain describe the force in objects undergoing. L b shearing stress on a cylinder deforming it by an angle. Request pdf deformation of solids introduce the concepts of irreversible. The elastic modulus is the constant of proportionality. The principal physical mechanism causing plastic deformation is sliding slip between planes of atoms in the crystal grains of the materials. The description of any deformation process involves specification of loads applied, which is the goal of a dynamic analysis. This internal resisting force per unit area is called stress. Stress strain relationship, hookes law, poissons ratio. When an external force acts on a body, it undergoes deformation.
We then develop a set of stress strain equations for a. The deformation corresponding to a 3d rigid rotation about an axis through the origin is yrx. Rigid and deformable bodies, strength, stiffness and stability stresses. En380 naval materials science and engineering course notes, u. Students are introduced to the concept of viscoelasticity and some of the material behaviors of viscoelastic materials, including strain rate dependence, stress relaxation, creep, hysteresis and preconditioning. Indeed, the propensity for rock deformation may be estimated using easily measured material properties such as the flow stress as a function of strain, strain rate and temperature, and applied to a large number of rocks. Solid surface stress is a fundamental property of solid interfaces. Strain can also be rotated to find its principal strain, principal strain direction, and maximum shear strain. To minimize deformation, select a material with a large elastic modulus e or g. A configuration is a set containing the positions of all particles of the body. Shear strain change in angle between two line segments originally perpendicular.
Mechanics of solids mechanics of solids finite deformation and strain tensors. Constitutive modeling of solids at finite deformation. These are sizeindependent measures of load and displacement, respectively. However, the external forces can change size and shape of any solid. In this paper, a deformation measure is introduced which leads to a class of strain measures in the lagrangian and eulerian descriptions. Me 160 introduction to finite element method chapter 4. Stress, strain and deformation in solids deformation mechanics. Most shapes occurring in the physical world are complex. The proportionality constant in this relation is known as the elastic modulus. Normal strain elongation or contraction of a line segment. For example, stress on a rubber band produces a larger strain as. Tensile, compressive and shear deformation of simple and compound bars under axial load, thermal stress. Stress is the measure of the force required to cause deformation.
Alevel physicsforces and motiondeformation of solids. Analysis of three dimensional stresses and strains. Stress is a directed force distributed over an area. Likewise, a small elastic modulus means that stress produces noticeable deformation. Strain is also a symmetric secondorder tensor, identical to the stress. We first construct a set of strain measures in terms of the x,y and z components of displacement at a point. For example, when a solid vertical bar is supporting an overhead weight, each particle in the bar pushes on the particles immediately below it. Permanent deformation of a sort that depends mainly on time of exposure to a stress and that tends to increase significantly with time of exposureis called viscous, or creep, deformation, and materials that exhibit those characteristics, as well as tendencies for elastic response, are called viscoelastic solids or sometimes viscoplastic. Mechanical deformation puts energy into a material.
The relation between stress and strain is given as follows. As the surface strain energy depends on the surface deformation gradient and its gradient, the kinetic energy is a quadratic function of the velocity and its surface. Mechanics of elastic solids lesson teachengineering. In order to develop a constitutive equation, a secondorder constitutive relation based on these strain measures is considered for modeling the mechanical behavior of solids at finite deformation. Deformation of solids stress and strain types of deformation. B plastic deformation occurs if, when the load is removed, the material contracts but a. Such classical nonlinear models are successful for calculating deformation in soft materials such as graphite, but can.
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