Advanced Mechanics of Materials and Applied Elasticity CHAPTER 1 INTRODUCTION - Overview - Analysis and Design - Definition and Components of Stress - Stresses on Inclined Sections - Principal Stresses and Maximum In-Plane Shear - Three Dimensional Stresses - Boundary Conditions in Terms of Surface Forces Overview - Solid Mechanics is a collection of physical laws, mathematical techniques, and algorithms that can be used to predict the behavior of any material subject to any type of loading. - Solid Mechanics is based on the following assumptions: ◦ Matter is infinitely divisible ◦ Matter is locally homogeneous; material properties are the same no matter how big or small your extraction is. ◦ Deformations are locally homogeneous ◦ The motion of any volume (or sub-volume) in a solid obeys Newton’s laws of motion ◦ Neighboring volume elements exert forces on each other that are equal and opposite. Overview Defining a Problem in Solid Mechanics ◦ Decide upon the goal of the problem and desired information. ◦ Identify the geometry of the solid to be modeled ◦ Determine the loading ◦ Decide what physics must be included; equilibrium equations ◦ Choose and calibrate a constitutive law that describes the behavior ◦ Choose a method of analysis ◦ Solve Decide Models are helpful, but you have to know what to model. ◦ Deformed shape of the component ◦ Forces required to cause particular shape change ◦ Stiffness of component, material properties ◦ Internal forces ◦ Critical forces (endurance limits, buckling limit, etc.) ◦ Natural Frequencies of vibration Vibration Deformation of a wine glass Decide (cont.) Solid mechanics principles are also used in a variety of failure mechanisms. For example, ◦ Predict the critical loads to cause a brittle or ductile fracture in a solid containing a crack (crack propagation) ◦ Predict the fatigue life of a component under cyclic loading ◦ Predict the rate of growth of a stress-corrosion crack in a component ◦ Predict the creep life of a component ◦ Find the critical crack length in a solid ◦ Predict the wear rate of a surface under contact loading ◦ Predict the fretting or contact fatigue life of a surface Decide (cont.) Furthermore, this science can be used in other applications besides just structural and mechanical analysis. For example, ◦ Calculating the material properties of a new material ◦ Predicting the influence of the microstructure on said mechanical properties ◦ Modeling the physics of failure in materials ◦ Modeling materials processing and heat treatments, and how those affect the aforementioned material properties as well ◦ Modeling biological phenomena and processes, such as bone growth and cell mobility Defining the Solid Sometimes, it’s not obvious how much of a true shape needs to be modeled. ◦ For brittle fractures, fatigue, or critical loads- all geometry should be modeled in great detail, as stress concentrations that are not apparent can be found. ◦ For creep damage, large scale plastic deformations, or vibrations, minute details are less important. ◦ Geometrical features often only influence local stresses ◦ So, as a rule of thumb, start with a simple model and see what it predicts. If it needs to be more detailed, then slowly add in more and more detail under you obtain a reasonable representation of what you’re trying to show. FEA model of a car crash Defining Loading There are six ways that mechanical loads can be introduced in a solid: ◦ The boundaries are subjected to a prescribed displacement or motion. (physical push/pull) ◦ The boundaries are subjected to a distribution of pressure or frictional traction (wind, frictional resistance) ◦ A boundary may be subjected to a combination of displacement and traction (physical push/frictional resistance) ◦ The interior of the solid can be subjected to gravitational or electromagnetic body forces (gravity) ◦ The solid can contact another solid (collisional force) ◦ Thermal expansion or material process that causes the solid to change shape (heat treatment, welding) Loading In can be surprisingly difficult in the industry to figure out exactly how to represent loads. Standards and Codes to the rescue! ◦ You can avoid excessive calculations if you define your critical limits as the ones dictated by whatever standard you’re designing to, as long as you have an adequate factor of safety. ◦ The trick is knowing what standards or codes your company/project is using, so be sure to obtain that information upfront. Physics ◦ Do you need to account for additional field quantities, such as temperature, electric/magnetic fields, or mass/solid diffusion through the solid? Temperature is the most common additional field quantity. ◦ The stress induced by temperature variation in a component is described as: ◦ 𝜎 = 𝐸 (𝛼𝑇)𝑚𝑎𝑥 −(𝛼𝑇)𝑚𝑖𝑛 ◦ Transient Heat conduction and plastic flow have similar formulas associated with them that, if needed, we will cover later. ◦ Is your analysis going to be static or dynamic? ◦ Are you solving a coupled fluid/solid interaction problem? ◦ Seen more in aerospace, offshore structures, pipelines, fluid containers. A coupled fluid-solid analysis of a wind turbine Defining Material Behavior How is your material supposed to behave? ◦ Isotropic linear elasticity- standard 𝜎 = 𝐸𝜖 behavior. Describes polycrystalline metals, ceramics, glasses, etc. Only two material properties are needed to describe material. ◦ Anisotropic linear elasticity- similar to the above, but materials are usually stiffer in some directions than others. Describes reinforced composites, wood, etc. Need between 3 to 21 material properties to characterize. ◦ Hyperelasticity- typically used to model rubber and foams, with large, reversible shape changes. Vastly complex; requires several parameters to obtain an accurate model. There are several other types of behavior that are not listed here, but the above list covers the few common ones. Analysis Method Once you have set up the problem, you will need to solve the equations of motion of your solid body to determine the stresses and strains in the solid. There’s a handful of ways to do this. ◦ 2D linear elastic solids: transforms, stress functions, complex variable methods ◦ 3D elastic problems ◦ 2D plane strain deformations But quite frankly, the ones that you will probably face more often than not, will be solved through: - FEA - Finite difference methods (similar to FEA) - Boundary equations - Free volume methods Analysis and Design Three pillars of Mechanical Design (Basic Principles of Analysis) ◦ Equilibrium conditions ◦ Material Behavior ◦ Geometry of Deformation Conditions of Equilibrium ◦ 𝐹 = 0, 𝑀=0 ◦ Statically determinate: all forces on its members can be found by only using the conditions of equilibrium. ◦ Statically indeterminate: more unknowns than available equations of statics. A Stressful Overview A body in equilibrium subject to a system of external forces will develop internal forces within the body. ∆𝐹𝑥 𝑑𝐹𝑥 𝜎𝑥 = lim = ∆𝐴→0 ∆𝐴 𝑑𝐴 ∆𝐹𝑦 𝑑𝐹𝑦 𝜏𝑥𝑦 = lim = ∆𝐴→0 ∆𝐴 𝑑𝐴 ∆𝐹𝑧 𝑑𝐹𝑧 𝜏𝑥𝑧 = lim = ∆𝐴→0 ∆𝐴 𝑑𝐴 Principal Stresses and Directions For any stress measure, we can associate a principal stress and direction as follows: Principal Stress: 𝜎𝑚𝑎𝑥,𝑚𝑖𝑛 = 𝜎1,2 = 𝜎𝑥 +𝜎𝑦 2 ± ( 𝜎𝑥 −𝜎𝑦 2 2 ) +𝜏𝑥𝑦 2 2𝜏𝑥𝑦 Principal Direction: tan 2𝜃𝑝 = 𝜎 𝑥 −𝜎𝑦 ◦ This will yield for you two directions, mutually perpendicular. One will correspond to the max stress, the other the min. Furthermore, we can discuss the maximum shearing stress and its direction too. Maximum Shearing Stress The maximum shearing stress can be found by following equation: 𝜏𝑚𝑎𝑥 = ± ( 𝜎𝑥 −𝜎𝑦 2 2 ) +𝜏𝑥𝑦 = 2 1 2 ± (𝜎1 − 𝜎2 ) Its direction can be found by: tan 2𝜃𝑠 = − obtained for 𝜃𝑝 . 𝜎𝑥 −𝜎𝑦 2𝜏𝑥𝑦 , or even more easily, adding 45° to the values Example 1.3 At a point in a structural member, the stresses are represented by figure 1.16a. Employ Mohr’s Circle to determine A) The magnitude and orientation of the principal stresses, and B) The magnitude and orientation of the maximum shearing stresses. Solution A) The principal stresses are represented by points 𝐴1 𝑎𝑛𝑑 𝐵1 . Therefore, the max and min stresses are given by: 𝜎1,2 = 60 ± 1 (80 4 − 40)2 +302 ; 𝜎1 = 96.05𝑀𝑃𝑎, 𝜎2 = 23.95𝑀𝑃𝑎 30 The planes are given by: 2𝜃𝑝 = tan−1 20 = 56.30°, 𝑎𝑛𝑑 56.30 + 180 = 236.30° Thus, 𝜃𝑝 = 56.30 2 (2) = 28.15°, 𝑎𝑛𝑑 𝜃𝑝 = 118.15° Solution B) The shearing stress is given by points D and E on Mohr’s Circle. The result is found as such: 𝜏𝑚𝑎𝑥 = ± 1 (80 4 − 40)2 +(30)2 = ±36.05 𝑀𝑃𝑎 The planes are found at 𝜃𝑝 + 45 = 73.15° 𝑎𝑛𝑑 163.15°. Three-Dimensional Stress Transformation cos 𝛼 = cos 𝒏, 𝑥 = 𝑙 cos 𝛽 = cos 𝒏, 𝑦 = 𝑚 cos 𝛾 = cos 𝒏, 𝑧 = 𝑛 𝑙 2 + 𝑚2 + 𝑛2 = 1 𝑝𝑥 = 𝜎𝑥 𝑙 + 𝜏𝑥𝑦 𝑚 + 𝜏𝑥𝑧 𝑛 𝑝𝑦 = 𝜏𝑥𝑦 𝑙 + 𝜎𝑦 𝑚 + 𝜏𝑦𝑧 𝑛 𝑝𝑧 = 𝜏𝑥𝑧 𝑙 + 𝜏𝑦𝑧 𝑚 + 𝜎𝑧 𝑛 Three-Dimensional Principal Stress Just like in the 2D case, 3D elements have three planes of zero shear that are mutually perpendicular and contain normal stresses that are the maximum and minimum stresses for the component. After derivation: 𝜎𝑥 − 𝜎𝑝 𝜏𝑦𝑥 𝜏𝑧𝑥 𝜏𝑥𝑦 𝜎𝑦 − 𝜎𝑝 𝜏𝑧𝑦 𝜏𝑥𝑧 𝜏𝑦𝑧 =0 𝜎𝑧 − 𝜎𝑝 3-D Stress (cont.) 𝜎𝑝3 − 𝐼1 𝜎𝑝2 − 𝐼2 𝜎𝑝 − 𝐼3 = 0, 𝐼1 = 𝜎𝑥 + 𝜎𝑦 + 𝜎𝑧 2 2 2 𝐼2 = 𝜎𝑥 𝜎𝑦 + 𝜎𝑥 𝜎𝑧 + 𝜎𝑦 𝜎𝑧 − 𝜏𝑥𝑦 − 𝜏𝑦𝑧 − 𝜏𝑥𝑧 𝜎𝑥 𝐼3 = 𝜏𝑦𝑥 𝜏𝑧𝑥 𝜏𝑥𝑦 𝜎𝑦 𝜏𝑧𝑦 𝜏𝑥𝑧 𝜏𝑦𝑧 𝜎𝑧 The Cauchy Stress Tensor 𝜎𝑥 𝜏𝑖𝑗 = 𝜏𝑦𝑥 𝜏𝑧𝑥 𝜏𝑥𝑦 𝜎𝑦 𝜏𝑧𝑦 𝜏𝑥𝑧 𝜏𝑦𝑧 𝜎𝑧 This is widely used in higher-level stress theory and is really the backbone of generic 3D Stress calculations. ◦ The principal stresses are the characteristic values, or eigenvalues. ◦ The directional cosines (l,m,n) are the eigenvectors. Example 1.7 The stress tensor in a machine element is given by the following array: 50 10 0 𝜏𝑖𝑗 = 10 20 40 Mpa 0 40 30 Determine the state of stress and 𝐼1 , 𝐼2 , 𝑎𝑛𝑑 𝐼3 by rotating x,y through an angle of 45° counterclockwise about the zaxis. Solution Through Eq. 1.28: 𝜏𝑖′𝑗′ 45 = −15 28.28 −15 25 28.28 28.28 28.28 𝑀𝑃𝑎 30 Also, it is seen through substitution that: 𝐼1 = 100𝑀𝑃𝑎, 𝐼2 = 1400 𝑀𝑃𝑎2 , 𝑎𝑛𝑑 𝐼3 = −53000 𝑀𝑃𝑎3 Example A rigid cubic solid is immersed in a fluid with mass density ρ. Recall that a stationary fluid exerts a compressive pressure of magnitude ρgh at depth h. - Write the expressions for the traction (stress) vector exerted by the fluid on each face of the cube. ◦ On the top face: t=ρgH𝑒3 ◦ On the bottom face: t= ρg(H+2a)𝑒3 ◦ On the side faces: t= -ρg(H+a-𝑥3) 𝑒1 , ρg(H+a-𝑥3) 𝑒1 , -ρg(H+a-𝑥3 ) 𝑒2 , ρg(H+a-𝑥3) 𝑒2 Example 2, Part A A rectangular bar is loaded in a state of uniaxial tension, as shown in the figure. - Write the components of the stress tensor in the bar, using the basis vectors shown. - The only non-zero component of stress on this rod is parallel to the 𝑒2 direction. 0 0 0 Thus: σ = 0 𝜎 0 0 0 0 Example 2, Part B Find the components of the normal vector to the plane ABCD shown, deducing the traction (stress) vector acting on this plane. - The normal vector is n=sin 𝜃𝑒1 + cos 𝜃𝑒2 - It follows that T= n∙ 𝜎 = sin 𝜃 - Thus, T= 0 𝜎 cos 𝜃 0 cos 𝜃 0 0 0 0 𝜎 0 0 0 0 0 Other Stress Measures The Cauchy Stress Tensor is the most widely used and understood measure of stress, but there are others out there. The bulk of these operate on the assumption that we regard forces acting on the undeformed solid, versus the deformed one. ◦ In laymen’s terms, Cauchy only cares about where you are, and these other measurements want to know where you are and where you came from. ◦ To use these, we need to know both what the deformed and undeformed solids look like. ◦ The deformation is described by a displacement vector u(x), and the associated deformation gradient, ▼. The general equation becomes: 𝐹 = 𝐼 + ∇𝑢, which can also be written as: 𝐹𝑖𝑗 = 𝛿𝑖𝑗 + 𝜕𝑢𝑖 𝜕𝑥𝑗 Other Stress Measures From these, we can define three other stress measures: ◦ Kirchhoff Stress: 𝜏 = 𝐽𝜎 ◦ Nominal Stress: 𝑆 = 𝐽𝐹 −1 ∙ 𝜎 ◦ Material Stress: = 𝐽𝐹 −1 ∙ 𝜎 ∙ 𝐹 −𝑇 ◦ These are somewhat useful in a general sense, but more often than not tend to overcomplicate the problems, thus I will mostly only hold you responsible for the Cauchy Stress Tensor, as that is the most practical application of this theory.
