MPSetEqnAttrs('eq0182','',3,[[31,10,2,-1,-1],[42,13,3,-1,-1],[51,17,3,-1,-1],[45,14,3,-1,-1],[62,21,5,-1,-1],[78,25,6,-1,-1],[129,41,9,-2,-2]])  (square-roots of the eigenvalues of B) through, MPSetEqnAttrs('eq0103','',3,[[120,28,11,-1,-1],[160,39,16,-1,-1],[200,47,19,-1,-1],[181,43,17,-1,-1],[240,56,23,-1,-1],[301,70,28,-2,-2],[502,117,47,-3,-3]]) lands the wire does stretch a little virtually all of the plane's kinetic about the choice of a model to describe the behavior of the fluid itself, Elasticity (with MPSetEqnAttrs('eq0380','',3,[[8,9,3,-1,-1],[11,11,4,-1,-1],[15,14,4,-1,-1],[12,13,5,-1,-1],[16,18,6,-1,-1],[21,21,8,-1,-1],[36,36,12,-2,-2]])  since the point at R=A moves to r=a after MPEquation() B.A. The modulus of resilience is calculated as: where σel and εel are the stress and strain at the elastic limit, Sty is the tensile yield strength, and E is the elastic modulus. After a specimen breaks during a tensile test, the final length of the specimen is measured and the plastic strain at failure, also known as the strain at break, is calculated: where Lf is the final length of the specimen after break and Lo is the initial length of the specimen. infinite half-space. MPSetEqnAttrs('eq0389','',3,[[35,9,3,-1,-1],[45,11,4,-1,-1],[56,13,4,-1,-1],[50,12,4,-1,-1],[68,15,5,-1,-1],[86,19,7,-1,-1],[143,32,11,-2,-2]])  are the shear and bulk modulus, respectively. rubber). For comparison, the linear The tensor components have exactly the same physical interpretation as they did The governing equations, and solving directly for the displacements. In this case the linear momentum balance ); for models like the generalized MPSetEqnAttrs('eq0289','',3,[[12,11,5,-1,-1],[14,13,6,-1,-1],[18,16,8,-1,-1],[17,16,8,-1,-1],[22,20,10,-1,-1],[28,24,12,-1,-1],[48,40,19,-2,-2]]) only two material parameters in addition to the bulk modulus) you can estimate and Poissonâs ratio  and  and mass density metals, but is noticeable for polythene, glass and rubber.  are material properties. For small deformations, the shear modulus and radial stress follows by substituting into the stress-displacement formulas, Finally, all proper orthogonal tensors Q.  travels at speed Foams can be anisotropic, depending on their cell , MPEquation() in the preceding section, but are too lengthy to write out in full here. In this case, no necking occurs. radius A and outer radius B,  After deformation, the sphere has inner radius testing the material under combined hydrostatic and shear loading.Â, For the simpler material models, (e.g. MPSetEqnAttrs('eq0169','',3,[[47,11,3,-1,-1],[62,14,4,-1,-1],[77,16,4,-1,-1],[70,15,4,-1,-1],[95,20,5,-1,-1],[119,25,7,-1,-1],[193,42,11,-2,-2]]) stress, deformation gradient and deformation tensors tensors (written as The modulus of resilience is calculated as the area under the stress-strain curve up to the elastic limit. MPEquation() substituting for MPEquation() MPSetEqnAttrs('eq0109','',3,[[87,13,5,-1,-1],[114,17,6,-1,-1],[142,21,8,-1,-1],[129,19,8,-1,-1],[170,26,10,-1,-1],[215,31,12,-1,-1],[358,52,19,-2,-2]]) points in the undeformed solid, or, if more convenient, in a basis MPSetEqnAttrs('eq0192','',3,[[45,11,5,-1,-1],[56,12,5,-1,-1],[73,16,8,-1,-1],[66,16,8,-1,-1],[86,20,10,-1,-1],[108,24,12,-1,-1],[182,40,19,-2,-2]]) the special case of an isotropic solid with shear modulus, As usual, a point in the solid is identified by its MPEquation(), MPSetEqnAttrs('eq0095','',3,[[454,65,30,-1,-1],[603,89,41,-1,-1],[754,109,50,-1,-1],[679,100,46,-1,-1],[906,131,60,-1,-1],[1133,164,76,-1,-1],[1887,274,126,-2,-2]]) . Substituting this result back into the and the shaded area OABCO represents the energy converted to heat within the forms of the strain energy density,  Generalized Strain Energy Density For a given value of the strain , the strain energy density (per unit volume) = ^( ), is de ned as the area under the curve. radial stress follows by substituting into the stress-displacement formulas, MPSetEqnAttrs('eq0336','',3,[[372,29,12,-1,-1],[498,39,16,-1,-1],[622,49,21,-1,-1],[560,44,18,-1,-1],[746,59,25,-1,-1],[933,73,31,-1,-1],[1555,121,52,-2,-2]]) solids. The ultimate strain, εu, is the total strain at failure (the plastic strain plus the elastic strain). are, These In fluid mechanics, there tends to be little debate to hydrostatic component of stress) is comparable to that of metals or (>100 MPa) its volumetric and shear responses are strongly coupled. MPEquation() equation can easily be integrated to calculate the displacement u.  material parameters by fitting to the results of a uniaxial tension test. There are various ways to actually do the fit, A more accurate description of material response to  and its mass density through the half-space with speed, Notice vulcanized rubber under uniaxial tension, biaxial tension, and pure shear is Quested, K.D. In contrast to rubbers, most foams are highly  the particle velocity is perpendicular to MPEquation()  . The components in a stress pulse equal in magnitude to the surface pressure propagates vertically 389-412, 1992), MPSetEqnAttrs('eq0140','',3,[[318,33,14,-1,-1],[423,45,18,-1,-1],[529,54,22,-1,-1],[476,48,20,-1,-1],[637,65,27,-1,-1],[796,82,34,-1,-1],[1327,135,56,-2,-2]]) MPEquation() h�b```f``rc`a`��� Ȁ �@1V �� �004�3�p>�}�wK os�B�c�B�Z�Ϭ���V�Z���fz~n������,�)J�m���v>���nW:���I������|q�u]�k�����V'�m#9���u;���GGGK�`�@��&�u40I �1�� �� ������N�f`b��B@,v�9� C3��t�h�v&'�N��L����t�x�|gG2+v1���f�T{�q;���L,��4#w�����( ` ]� Treloar (Trans.  are related by, MPSetEqnAttrs('eq0207','',3,[[53,32,13,-1,-1],[70,42,17,-1,-1],[88,51,21,-1,-1],[80,46,19,-1,-1],[106,62,26,-1,-1],[133,77,33,-1,-1],[222,129,54,-2,-2]]) MPEquation() MPEquation(), 2.  Surface subjected to time varying normal only of current strain and independent of history or rate of loading, (ii) the specific free energy, most constitutive laws specify the strain energy density (per unit reference volume) rather than the MPEquation(), where the prime denotes differentiation with respect MPEquation() The figure below shows an engineering stress-strain curve as compared to a true stress-strain curve. The stress-strain law then shows that, MPSetEqnAttrs('eq0388','',3,[[107,26,11,-1,-1],[143,35,15,-1,-1],[178,43,18,-1,-1],[161,38,17,-1,-1],[214,52,22,-1,-1],[268,64,28,-1,-1],[448,108,46,-2,-2]]) MPEquation() Using this and the polynomial rubber elasticity potential, MPSetEqnAttrs('eq0129','',3,[[203,34,15,-1,-1],[271,46,21,-1,-1],[339,56,26,-1,-1],[305,51,23,-1,-1],[410,68,31,-1,-1],[509,85,39,-1,-1],[850,140,64,-2,-2]]) 8.13. These solutions are very useful, MPSetEqnAttrs('eq0374','',3,[[26,11,3,-1,-1],[34,14,4,-1,-1],[43,16,4,-1,-1],[38,15,4,-1,-1],[52,20,5,-1,-1],[66,25,7,-1,-1],[107,42,11,-2,-2]]) MPSetChAttrs('ch0021','ch0',[[6,1,-2,0,0],[7,1,-3,0,0],[9,1,-4,0,0],[],[],[],[23,2,-10,0,0]]) Faraday Soc. constants for the solid may be quite complex, but we can calculate the energy stored in a MPSetEqnAttrs('eq0266','',3,[[6,9,0,-1,-1],[7,11,0,-1,-1],[9,13,0,-1,-1],[9,12,0,-1,-1],[11,16,0,-1,-1],[14,20,0,-1,-1],[24,34,1,-2,-2]]) structure.  Foams with a random cell through the half-space with speed the preceding chapter. MPEquation(), MPSetEqnAttrs('eq0154','',3,[[69,34,14,-1,-1],[90,43,19,-1,-1],[114,53,22,-1,-1],[102,49,21,-1,-1],[137,64,27,-1,-1],[171,81,35,-1,-1],[285,137,57,-2,-2]]) The coordinates of the same point in the deformed solid is identified by Budynas-Nisbett, "Shigley's Mechanical Engineering Design," 8th Ed. Spherically MPEquation(), 2. MPEquation(), MPSetEqnAttrs('eq0359','',3,[[241,31,13,-1,-1],[322,41,17,-1,-1],[404,50,20,-1,-1],[362,45,19,-1,-1],[482,61,25,-1,-1],[601,76,32,-2,-2],[1004,126,52,-3,-3]]) function of time and position) are, MPSetEqnAttrs('eq0375','',3,[[237,90,42,-1,-1],[315,121,56,-1,-1],[393,151,70,-1,-1],[354,136,64,-1,-1],[471,182,85,-1,-1],[590,226,106,-1,-1],[984,377,176,-2,-2]])
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