MECH 5312 Solid Mechanics II Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso
Table of Contents Thermodynamics Derivation Hooke s Law: Anisotropic Elasticity Hooke s Law: Isotropic Elasticity Thermoelasticity Hooke s Law: Orthotropic Materials
Thermodynamics Symbolically, the first law of thermodynamics is expressed by the equation W H U K where W is the work performed on the system by external forces, H is the heat that flows into the system, U is the increase in internal energy, and is the increase in kinetic energy K Let s consider a infinitesimal element of Volume, V
Thermodynamics The system is subject to infinitesimal increments (variations) in the displacement components (u, v, w) indicated by (δu, δv, δw). Subsequently, the variation of the strain components becomes xx yy zz u 1 u v, xy x 2 y x v 1 v w, yz y 2 z y w 1 w u, zx z 2 x z Notice, we are using the mathematical shear strain, ε xy 1 1 1 2 2 2 xy xy yz yz zx zx
Thermodynamics Let s consider an adiabatic system where the net heat flow into V is zero, δh=0 Let s consider static equilibrium where the kinetic energy is zero, δk=0 For V, the first law of thermodynamics regress to W H U K W U
Thermodynamics Variation of Work done on V is derived from Forces Forces can be divided into those acting over the surface and exerted on the Body. Consider a point P of surface ds A traction, t acts on ds with normal, n. t n Px Py Pz n n n n x y z W Fu
Thermodynamics Variation of work due to Surface forces f f s b Surface Force Vector S t ds t x, t Traction Vector ds Surface Increment Px n ( n) t Py t Pz S n S xx xy xz yx yy yz zx zy zz W uds vds wds S S Px Py Pz S S S xxnx yzny zxnz u xynx yyny zynz v xznx yzny zznz w ds
Thermodynamics Variation of work due to Body forces acting throughout V b f b dv m b dv mg f b V Force Vector x, t Density b x, t Body Force Vector x Current Configuration V bx b b y B b b z W B u B v B w dv B x y z V
Thermodynamics We can find the variation of total total work by adding the variation of Surface work and Body work together. W W W The surface integral can be converted to a volume integral via the divergence theorem. V u dv u nds x S B Eliminates the normal vector, n W xx yz zx u xy yy zy v xz yz zz w x y z dv V B xu By v Bz w
Thermodynamics We use equilibrium to eliminate the Body forces Introduce variation of strain components to find the variation of work xx yx zx Bx x y z 0 xy yy zy By x y z 0 xz yz zz Bz x y z 0 W W W 2 2 2 dv S B xx xx yy yy zz zz xy xy yz yz zx zx V
Thermodynamics Since, W U The external work is equal to the internal energy. The internal energy U for V is expressed in terms of the internal energy per unity volume. That is, in terms of the internal-energy density U 0. As a variation of internal energy, U U0dV V U0 2 2 2 xx xx yy yy zz zz xy xy xz xz yz yz
Elastic and Complementary Internal-Energy σ ε 0 0 VEC VEC U C VEC σ xx yy zz xy yz xz yx zy zx VEC ε xx yy zz xy yz xz yx zy zx
Elasticity and Internal-Energy Density The strain-energy density, U 0 is a function of certain variables; we need to determine those variables. For linear elastic homogeneous materials, the total internal energy, U in a loaded member is equal to the potential energy of the internal forces (called elastic strain energy).
Elasticity and Internal-Energy Density The internal energy is a function of strain and temperature As a variation it can be formed as Stress components arises as U0 U0,,,,,, T xx yy zz xy xz yz U U U U U U U 0 0 0 0 0 0 0 xx yy zz xy xz yz xx yy zz xy xz yz U U U 0 0 0 xx yy zz xx yy zz 1 U 1 U 1 U 0 0 0 xy xz yz 2 xy 2 xz 2 yz
Hooke s Law: Anisotropic For 1D linear-elastic homogeneous materials we know Hooke s Law as E Strain Energy as U 0 1 2 In reality,
Hooke s Law: Anisotropic Many classes of materials exhibit a linear elastic response including: metals, concrete, wood, etc. These continua are also called Hookean Solids because they obey Hooke s law. Generalized Hooke s Law C S S C 1 ij ijkl kl ij ijkl kl ijkl ijkl C S ij ij ijkl ijkl 2nd order stress tensor 9 terms 2nd order strain tensor 9 terms 4th order stiffness tensor 81 terms 4th order compliance tensor 81 terms
Hooke s Law: Anisotropic In this form, there are 9 equations and 81 stiffness terms to find. C C... C 11 1111 11 1112 12 1133 33 C C... C 12 1211 11 1212 12 1233 33 C C... C 33 3311 11 3312 12 3333 33
Hooke s Law: Anisotropic Let us take advantage of the symmetric stress and strain tensor arrived in previous chapters Such that, ij ji ij ji C C C C ijkl jikl ijkl ijlk This symmetry in the Stiffness Tensor reduces the number of intendent terms to 36. For most materials the number of independent constants is much lower. This reduction is due to material symmetry.
Hooke s Law: Anisotropic Hooke s law resolves into a system of equations where C ij are 36 elastic coefficients and Cij Cji
Hooke s Law: Anisotropic Let s evaluate the following 9 partial derivatives
Hooke s Law: Anisotropic Perform the appropriate double partial derivatives to find This reduces the number of district C ij coefficients to 21
Hooke s Law: Anisotropic Introducing these partial derivative together, U U U U U U U 0 0 0 0 0 0 0 xx yy zz xy xz yz xx yy zz xy xz yz The strain energy density resolves to 1 2 1 1 U0 C11 xx C12 xx yy C16 xx yz 2 2 2 1 1 2 1 C12 xx yy C22 yy C26 yy yz 2 2 2 1 1 1 C13 xx zz C23 yy zz C36 zz yz 2 2 2 1 1 1 C C C 2 2 2 2 16 xx yz 26 yy yz 66 yz
Hooke s Law: Isotropic Elasticity Let s consider an isotropic material. Random Oriented Grains Homogenize over a Representative Volume Element. Material symmetry is exists in all directions Only two coefficients are required to characterize the Hookean response
Hooke s Law: Isotropic Elasticity In the principal direction, the strain energy density is For isotropic materials, this regresses to where only two coefficients, Lame s elastic coefficients, λ and G exist.
Hooke s Law: Isotropic Elasticity In terms of general strain The 3D Hooke s Law in terms of Lame which is easy to write. e xx yy zz
Hooke s Law: Isotropic Elasticity In practice, the Lame coefficient λ is not easy to find because it has no physical meaning. As an alternative we introduce Young s Modulus and Poisson s Ratio These constants arise in a specimen subject to uniaxial tension xx E xx, yy xx
Hooke s Law: Isotropic Elasticity 3D Hooke s Law in terms of Young s Modulus and Poisson s Ratio which is not so easy to write!
Hooke s Law: Isotropic Elasticity Plane Stress is the case of thin members where the state of stress can be simplified to 2D
Hooke s Law: Isotropic Elasticity Plane Strain is the case of thick members where the state of strain can be simplified to 2D
Hooke s Law: Isotropic Elasticity Strain energy density in terms of Young s modulus and Poisson's ratio
Hooke s Law: Isotropic Elasticity Young s/poisson to Lame Lame to Young s/poisson Bulk Modulus Where K relates the mean stress to the dilitation
Example Solecki P4-4, Boresi P3-5, P3-10
Thermoelasticity Consider an unconstrained member made of an isotropic material The member is subject to a temperature change T Thermal strain develops in the member as follows xx, yy, zz Uniform Volume Expansion T 0 xx yy zz xy yz xz where α is the thermal expansion coefficient.
Thermoelasticity Now consider the same member but subject to forces as well as a temperature change. xx, yy, zz, xy, yz, xz Mechanical Strains due to Force xx, yy, zz Uniform Volume Expansion
Thermoelasticity Total strain, ε ε ε T I M Mechanical strain, ε M arises from hooke s Law ε M Cσ Stress due to thermal and mechanical strain T T 1 σ C ε I S ε I
Thermoelasticity Stress due to thermal and mechanical strain Not easy to express in Lame form.
Thermoelasticity Together with Hooke s Law The equation for strain energy density, U 0 does not change but the temperature distribution may effect the stresses.
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
Hooke s Law: Orthotropic Materials 3D Hooke s in terms of Young s modulus and Poisson s ratio
Example Boresi Ex 3.5
Hierarchal Materials
Contact Information Calvin M. Stewart Assistant Professor Department of Mechanical Engineering The University of Texas at El Paso 500 W. University Blvd. Suite A126 El Paso, Texas 79968-0717 Email: cmstewart@utep.edu URL: http://me.utep.edu/cmstewart/ Phone: 915-747-6179 Fax: 915-747-5019