2.3 Basic Continuum Mechanics

Transcription

1 Chem 554/Overney. Basic Continuum Mechanics Index.. lastic Moduli and Free nergy Relations..... Special cases of elasticity and methods..... Cantilever lasticity, Sample and Contact Stiffness lastic Moduli and Free nergy Relations The elastic concept of continuum mechanics starts with Hooke s law which linearly relates the stress ( or τ ) and the strain ( or γ ) of an ideal elastic body. In the case of an uniaxial elongation/compression in x-direction or simple shear in y-direction of an isotropic material, Hooke s law has the following simple form: xx xx. ( a) τ Gγ. ( b) xy with the modulus of elasticity (Young s modulus) and the shear modulus G. The left and right indices of the stress and the shear denote the normal vector of the shear plane and the direction of the force, respectively. The dimensions of the stress is a force per unit area (i.e., a pressure) while the strain is dimensionless - a relative measure of the local displacement u. The strain is defined for symmetry reason as u u i j + i x j x ;,,. () i If is independent of the location r, the deformation can be expressed as u*r. The index i represents the normal vector of the plane in respect to which the deformation occurs. The index j determines the direction of the deformation. In the case of a hydrostatic compression, the non-zero-components of the stress tensor correspond to a constant pressure p, i.e. -pδ. The specific work (i.e., the work per volume element) resulting from internal strain is δw δ () where δ represents the changes in the strain tensor. Internal strain in a body that results from external forces ceases to exist if the external forces are removed and the body behaves fully elastic. If the body is plastically deformed, a partial strain remains. Deformations which are slow enough, i.e., the body is in thermal equilibrium at all xy A vector is denoted in bold face and a tensor is doubled underlined. δ is the Kronecker symbol.

2 Chem 554/Overney times, are reversible. In the case of a reversible deformation, the differential of the internal energy per volume element is de Tds + d (4) with T as the absolute temperature, ds as the entropy per volume element, and Tds the heat involved during the deformation. Hence, the free energy F-TS can expressed as df sdt + d (5) where f is the free energy per volume element. Defining the enthalpy as h e Ts f u (6) the following thermodynamic relationships are found: e f ; (7a) s const. T const. h T const.. (7b) The free energy has to be expressed as a function of the shear tensor in order to apply the thermodynamic relationships. For very small deformation the general expression for the free energy of a deformed isotropic body is λ f f o + ii + µ (8) with the Lamé coefficient λ and µ. Any deformation can be written as the sum of pure shear and homogeneous dilatation by δ kk + δ kk. (9) The free energy can then be written as follows: K f fo + µ δkk + kk (0) where µ is the modulus of rigidity (also called shear modulus) and Kλ+/µ is the modulus of compression. Both moduli should be larger than zero. Hence, the stress tensor for a small deformation of an isotropic solid body can be expressed as the sum of a dilatation and pure shear component as f Kkkδ µ δkk +. (a) T const. The reversed expression is δ µ δ kk kk K +. (b) 9 Tensor algebra (instein relation): ii + + and ik,.

3 Chem 554/Overney In the case of a uniaxial deformation in x direction (i.e., 0 and ii 0 for i,), equation (b) reduces to µ +. () K which leads with the equation (a) to the fundamental definition of the Young s modulus Kµ + µ λ µ 9. () K + µ λ + µ The shear modulus µ (to avoid any conflict with the coefficient of friction also called G modulus) can be expressed as µ G ( + ν). (4) where ν is the ratio between the deformation parallel to the applied force and the deformation perpendicular to the applied load - also known as Poisson s ratio. 4 The modulus of compression K is K ( υ ). (5) Finally the specific free energy can be written as f fo + + kk ( + ν) υ υ (6).. Special cases of elasticity and methods Thermodynamically ν is only restricted to [-,½] which would allow the body to extend its cross-sectional area during longitudinal extension. To avoid that crosssectional area extension it suggests restricting the regime of ν further to [0,½]. The stress and the shear tensors are expressed by the Young s modulus and the Poisson s ratio for a small longitudinal deformation as υ + υ ν δ, (7a) + kk and (( + ν) νkkδ), (7b) respectively. The volume change of bulk polymers during compression or tension is in most cases very small in comparison to the deformation. Hence, ν ½ for polymers which simplifies the Young s modulus - shear modulus relation to G. In the case of a plane stress assumption (used, for example, in thin plates), where the stress components perpendicular to the plane of the film, and 4 ν K µ λ. K + µ ( λ + µ )

4 Chem 554/Overney are neglected, the two-dimensional stress-strain relation can be obtained from equations (7) as υ 0 υ 0, + ν 0 0 (8a) ν ( ) ν +, (8b). (8c) 0 An anisotropic material has independent elastic constants. Once the orthotropic axes of symmetry are known, the number of elastic constants needed to fully characterize the material reduces to 9. Orthotropic materials are unidirectional composites, woods, and laminated metallic products. With the principal axes of orthotropy as the reference axes, the strain-stress relation can be written as: υ υ υ υ υ υ υ G G Five coefficients characterize the material,,, G, ν and ν. There are various techniques that are used to determine elastic constants of material. The techniques can be grouped in roughly three groups: (i) static tensioncompression or bending tests, (ii) vibration tests, and (iii) wave propagation tests. Static measurements are technically easy to conduct and provide information about the extensional moduli, the shear moduli and the Poisson s ratios. An example of the static method is the cantilever beam method, Figure 4.. The cantilever beam, the substrate, is coated with a film, the sample. The thickness of the film t f is assumed to be much smaller than substrate thickness t s. Further, it is assumed that the distortion at the end of the beam δ is smaller than the sample film thickness. Based on this assumption the following relation is used to determine the stress f in the film which acts along the cantilever beam: sδ ts ftf, (0) ( νs) L where s and ν s are the Young s modulus and Poisson s ratio of the substrate. (9) 4

5 Chem 554/Overney Figure 4.: Cantilever beam method used to measure in-plane stresses of ultrathin films. X-ray and electron diffraction methods are applied to determine in-plane stresses of ultrathin films supported by solid substrates using the plane stress assumption from above and assuming in-plane isotropic deformation, i.e., and. These methods work for crystalline material and are sensitive to variations in the lattice parameters. For a crystal, stress and strain are related by kl c (a) kl s (b) with the elastic constants and coefficients (compliances) c kl and s kl, respectively. For films of isotropic elastic behavior, it follows from equation (8) that (ν-)/(ν). The Poisson s ratios, ν and ν, of an orthotropic ultrathin polymeric material can be determined using vibrational holographic interferometry (). The tested polymer film resembles a vibrating membrane and its residual stresses are measured and analytically analyzed by the wave equation for a vibrating membrane. Principle stresses are measured for D constrained and D constrained films. Poisson s ratios are determined from D D ν D. (a) ν kl D kl D D (b) Other methods that are used to establish elastic constants of materials are, for instance, tensile testing, high pressure gas dilatometer, pressure-volume-temperature apparatus, torsion pendulum and scanning force microscopy. Before we, however, further discuss techniques another material property, besides elasticity, has to be introduced - viscosity. Source: Landau Lifshitz, Mechanics 5

6 Chem 554/Overney.. Cantilever lasticity, Sample and Contact Stiffness First of all, one has to differentiate between two scales of stiffness: (i) the elastic modulus of the material (expression for the material stiffness, and (ii) the device stiffness, which is typically expressed as a spring constant, k. The Young's modulus,, and the shear modulus, G are the most common examples for elastic moduli. Considering the fully elastic deformation caused by a spherical indenter tip on the surface of a half space, the deformation defines the contact stiffness, k c, which, based on Hertz Theory, is related to the combined Young's modulus of the tip and half space, as * v v kc a with * + where i and ν i (I,) the material Young's moduli and Poisson Ratios. The contact stiffness is sometimes also referred to as sample stiffness. The force or "normal load applied", F N, measured in the process of an indentation can be expressed as FN keff z kc zs kl zl with z z s + zl, where z is the combined compression, measured in z-direction, of the indenter (cantilever) spring, z L, and the surface of the half space, z s. Consequently, the effective spring constant, k eff, can be expressed as k eff + kc k. L For a bar-shaped cantilever with length L, width W and thickness t, and an integrated tip of length r, the normal and torsional spring constants, k N and k t, are related to the material stiffnesses, as Wt 4L GWt k t. Lr k N and The thickness of the cantilever, typically poorly defined by the manufacturers, can be determined from the first resonance frequency of the "free" cantilever using the following empirical equation: t πf ρ L ( ) The Young's modulus and density of silicon cantilevers are around.69 0 N/m and ρ. 0 kg/m. Nanoscience Friction and Rheology on the Nanometer Scale,. Meyer, R.M. Overney et al., World Scientific, NJ (998). 6