Linear Stress Strain Temperature Relations. 3.1 First Law of Thermodynamics, Internal-Energy Density, and Complementary Internal-Energy Density

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1 hapter 3 Linear Stress Strain Temperature Relations 3.1 First Law of Thermodynamics, Internal-nergy Density, and omplementary Internal-nergy Density If electromagnetic effects are disregarded, the first law of thermodynamics is described as follows: The work performed on a mechanical system by external forces plus the heat that flows into the system from the outside equals the increase in internal energy plus the increase in kinetic energy. δ W δh δu + δk + (3.1) 115

2 3.2 Hooke s Law: Anisotropic lasticity, In the one-dimensional case, for a linear elastic material where the proportionality factor is called the modulus of elasticity. The relation is Hooke s law. More generally, in the three-dimensional case, Hooke s law asserts that each of the stress components is a linear function of the components of the strain tensor (3.2) 116

3 These are examples of composite materials that have different properties in different directions (anisotropic materials). Such materials are characterized by 3-D Hookean relationships. A cross section view of a carbon-epo composite showing the strong and stiff graphite fibers embedded in the tough epo matrix (By B. D.Agrawal, L. J.Broutman and K. handrashekhara. Analysis and performance of Fiber composites) 117

4 xample 3.1: Bending of a plate around a cylinder A flat rectangular plate lies in the (x,y) plane. The plate, of uniform thickness h 2.mm, is bent around a circular cylinder with the y axis parallel to the axis of the cylinder. The plate is made of an isotropic aluminum alloy ( 72. GPa and v.33). The radius of the cylinder is 6mm. (a) Assuming that plane sections for the undeformed plate remain plane after deformation, determine the maximum circumferential stress in the plate for linearly elastic behavior. θθ ( max) Note: θθ. rr ( v v ) ( v v ) ( v v ) rr rr rr (3.3), There is no shear. 118

5 xamples of cylindrical pressure vessels (production on the left, pressure vessel on the right). Although the vessels are welded, sheet metal is first bent into a cylindrical shape. From: 4 roll Full Hydraulic plate roll AD-AM DN "Pro2" Applications: Rolling any shape of plate (cylindrical, conical, oblong, oval, multi-radii) to manufacture pipes, tanks, boilers, shovels, and any kind of round shaped shell. From: 119

6 3.4 quations of Thermoelasticity for Isotropic Materials onsider an unconstrained member made of an isotropic elastic material. Let the uniform temperature of the member be increased by a small amount T. xperimental observation has shown that, for a homogeneous and isotropic material, all infinitesimal line elements undergo equal expansions and maintain their initial directions (no shear). Therefore, the strain components resulting from the temperature change are T, α T (3.34) α where denotes the coefficient of thermal expansion of the material. Now let the member be subjected to forces that induce stresses,,,. If,,, denote the total strain components after the application of the forces, the change in strain produced by the forces is 12

7 , α T,, α T, α T (3.35) The generalized form of 3.3 is [ v( + )] ( + ) [ v ] [ v( + )] ( 1+ v) ( 1+ v) ( 1+ v), + α T + α T + α T, (3.38) 121

8 The strain energy density is U 1 2 λ ( ) 2 ( G ) c 3 2 ( + + ) T + cα( T ) 2 (3.4) c α ( 1 2ν ) Note: Temperature affects all material constants, including the moduli of elasticity and shear, T, and the Poisson ratio (the latter is usually little affected). This makes the stress-strain-temperature relationships (3.38) much more complicated to use. The relevant data on the effect of temperature on properties is sometime provided by material suppliers, but in general, it is not easy to find. ngineers dealing with thermal problems should be careful!! 122

9 xample: ffect of temperature on the modulus of elasticity of metals. From: 123

10 3.5 Hooke s Law: Orthotropic Materials Materials such as wood, laminated plastics, cold rolled steels, reinforced concrete, various composite materials, and even forgings can be treated as orthotropic. Orthotropic materials possess three orthogonal planes of material symmetry and three corresponding orthogonal axes called the orthotropic axes. If x, y, z are the orthotropic axes with the planes of material symmetry (x, y), (y, z), (x, z), such that properties are symmetric about each of these planes, the elastic coefficients ij are (3.49) 124

11 Tail of a helicopter made of carbon-fiber reinforced plastic ontinuing the 45 year development of the famous aircraft platform, the next-generation Learjet will be the first Bombardier Aerospace jet to feature an all-composite structure as well as the first all-composite structure business jet designed for type certification under FAR Part 25. Launched in October /8761/picture/4272/ 125

12 Other composite designs From: From A fiber-reinforced material has higher strength and stiffness in the fiber direction, while its stiffness and strength in the plane perpendicular to the fibers are identical in all directions. 126

13 Therefore, there are 9 independent elastic coefficients. The stress-strain relations become (3.5) The stress-strain relations for orthotropic materials in terms of orthotropic moduli of elasticity and orthotropic Poisson s ratios can be written in the form 127

14 xample 3.6: Stress-Strain Relationship in Fiber-Reinforced omposites (micromechanical aspects are included, i.e. we know the properties of fibers and the matrix and the goal is to formulate the stress-strain relations for the composite material) A lamina (a thin plate, sheet, or layer of material) of a section of an airplane wing is composed of unidirectional fibers and a resin matrix that bonds the fibers. Let the fiber volume fraction (the proportion of fiber volume to the total volume of the composite) be f. Determine the effective linear stress-strain relations of the lamina. Porosity that is often present is neglected! Figure 3.6 Lamina: fiber volume fraction f, resin volume fraction 1 f. 128

15 mineral aggrecan collagen fiber collagen fiber mineral Tendon-to-bone insertion site: collagen fibers in tendons. From: Washington U. and MST (Victor Birman) research. 129

16 xample 3.7: Thin-walled omposite ylindrical Shell Subject to Axisymmetric Mechanical and Thermal Loading (stress analysis) onsider a composite cylinder of length L formed from an inner cylinder of aluminum with outer radius R and thickness t A and an outer cylinder of steel with inner radius R and thickness t S (Figure 3.7a); t A << R and t S << R. The composite cylinder is supported in an upright, unstressed state between rigid supports. An inner pressure p is applied to the cylinder (Figure 3.7b), and the entire assembly is subjected to a uniform temperature change T. Determine the stresses in both the aluminum and the steel cylinders for the case t A t S t.2r. For aluminum, A 69 6 GPa, v A.333, and α For steel, S 27 GPa, A /. 6 v S.28, and α S /. Note: Displacements of the end cross sections are prevented, i.e. the cylinder cannot expand in the axial direction. 13

17 Pressure vessel produced by Fluitron, Inc. Pressure vessel with spherical caps 131