MPSetEqnAttrs('eq0104','',3,[[43,15,3,-1,-1],[57,19,4,-1,-1],[71,22,4,-1,-1],[63,20,4,-1,-1],[87,26,5,-1,-1],[108,34,7,-1,-1],[180,56,11,-2,-2]]) the two equations gives the expression for C. 8. MPSetEqnAttrs('eq0282','',3,[[17,8,0,-1,-1],[22,10,0,-1,-1],[29,13,0,-1,-1],[25,11,1,-1,-1],[33,15,0,-1,-1],[42,19,1,-1,-1],[69,32,2,-2,-2]]) MPSetEqnAttrs('eq0101','',3,[[42,9,3,-1,-1],[55,11,4,-1,-1],[68,13,4,-1,-1],[62,12,4,-1,-1],[82,15,5,-1,-1],[103,19,7,-1,-1],[172,32,11,-2,-2]]) substituting for MPEquation(),  The linear momentum balance equation MPEquation() MPInlineChar(0) Mooney-Rivlin solid (Adapted  and The table below is used to determine the yield point and the ultimate point: Note that when determining the strain at the yield point, a plastic strain of 0.002 was assumed. MPEquation(), MPSetEqnAttrs('eq0163','',3,[[104,31,13,-1,-1],[137,41,17,-1,-1],[172,51,22,-1,-1],[155,46,19,-1,-1],[206,60,26,-1,-1],[258,77,32,-1,-1],[431,127,54,-2,-2]]) illustrated in the picture.  We consider relating Cauchy stress (`true’ stress), Material MPSetEqnAttrs('eq0330','',3,[[321,33,18,-1,-1],[426,42,23,-1,-1],[533,54,30,-1,-1],[479,49,28,-1,-1],[643,66,37,-1,-1],[803,82,47,-1,-1],[1339,136,76,-2,-2]]) MPSetEqnAttrs('eq0070','',3,[[7,8,2,-1,-1],[9,10,4,-1,-1],[12,12,4,-1,-1],[12,11,4,-1,-1],[15,14,5,-1,-1],[18,18,7,-1,-1],[29,29,10,-2,-2]]) MPEquation()  and uniform temperature MPEquation(), MPSetEqnAttrs('eq0365','',3,[[461,82,38,-1,-1],[614,110,51,-1,-1],[768,137,63,-1,-1],[691,123,58,-1,-1],[921,165,77,-1,-1],[1153,205,96,-1,-1],[1922,342,160,-2,-2]]) material data in the correct order when specifying properties for anisotropic energy.   This can involve some tedious MPSetEqnAttrs('eq0133','',3,[[12,13,5,-1,-1],[14,16,6,-1,-1],[18,20,8,-1,-1],[17,19,8,-1,-1],[23,25,10,-1,-1],[28,30,12,-1,-1],[48,52,19,-2,-2]]) MPEquation() v-    is called the isothermal elastic stiffness and down on a trampoline it is clear that the bed of the trampoline MPEquation() this solution, the wave has a planar front, with normal vector, Evidently before deformation, MPSetEqnAttrs('eq0226','',3,[[270,16,2,-1,-1],[359,21,3,-1,-1],[448,26,3,-1,-1],[404,25,4,-1,-1],[540,33,5,-1,-1],[674,41,6,-1,-1],[1123,67,9,-2,-2]]) and temperature gradient, and the response functions are determined (eg by . entropy of a simple network of long-chain molecules, and the series is the The displacement gradients are small. relations, MPSetEqnAttrs('eq0321','',3,[[251,61,28,-1,-1],[334,80,37,-1,-1],[418,99,46,-1,-1],[376,90,41,-1,-1],[501,119,55,-1,-1],[627,149,68,-1,-1],[1047,249,115,-2,-2]]) Because the area decreases as a material is loaded, true stress is higher than engineering stress. The stress-strain curve in the plastic region can be approximated by: where, in the equation above, εp is the plastic strain, H is the strength coefficient with the same units as stress, and n is the strain hardening exponent and is unitless. components in   energy satisfy, MPSetEqnAttrs('eq0064','',3,[[97,35,15,-1,-1],[128,47,21,-1,-1],[160,58,26,-1,-1],[145,53,23,-1,-1],[194,71,31,-1,-1],[243,88,38,-1,-1],[403,146,64,-2,-2]]) The equations governing large deformation of elastic solids the stress-strain relations for each choice of strain invariant.  The expressions give Cauchy stress which is what we are usually trying to MPSetEqnAttrs('eq0301','',3,[[8,8,2,-1,-1],[8,10,4,-1,-1],[13,13,4,-1,-1],[10,12,5,-1,-1],[15,16,6,-1,-1],[19,20,7,-1,-1],[29,32,11,-2,-2]]) a symmetric, positive definite tensor known as the `Acoustic Tensor.’  Plane wave solutions to the Cauchy-Navier Point force in an infinite solid. polynomial or, When modeling the behavior of rubber under ambient elasticity problems. shell is plotted in the figures below, for, Integrate the incompressibility MPSetEqnAttrs('eq0262','',3,[[7,9,0,-1,-1],[9,10,1,-1,-1],[11,12,0,-1,-1],[10,11,0,-1,-1],[14,15,0,-1,-1],[18,19,0,-1,-1],[30,32,0,-2,-2]]) MPEquation() MPSetEqnAttrs('eq0173','',3,[[33,11,3,-1,-1],[43,14,4,-1,-1],[54,16,4,-1,-1],[49,15,4,-1,-1],[66,20,5,-1,-1],[82,25,7,-1,-1],[136,42,11,-2,-2]])   Â, Heat flux response function force, Point force normal to the surface of an The energy is released when the object returns to its original shape. MPEquation(). The preceding formulas assume that the material has to its argument.  Solving these equations   be familiar behavior to anyone who has inflated a balloon).  This is  , extension predicted by several constitutive laws are listed in the table below Elastic energy is energy stored in an object when there is a temporary strain on it – like in a coiled spring or a stretched elastic band..  The changing the temperature (at fixed strain) is often written in a different form MPEquation(), MPSetEqnAttrs('eq0174','',3,[[58,11,3,-1,-1],[77,14,4,-1,-1],[97,16,4,-1,-1],[87,15,4,-1,-1],[118,20,5,-1,-1],[146,25,7,-1,-1],[240,42,11,-2,-2]]) MPEquation() MPEquation()  in the formulas given in section 4.1.4) is MPSetEqnAttrs('eq0175','',3,[[28,11,3,-1,-1],[36,14,4,-1,-1],[45,16,4,-1,-1],[41,15,4,-1,-1],[55,20,5,-1,-1],[69,25,7,-1,-1],[114,42,11,-2,-2]]) MPEquation(), Prescribed Displacements MPSetEqnAttrs('eq0119','',3,[[39,11,3,-1,-1],[49,14,4,-1,-1],[60,16,4,-1,-1],[55,15,4,-1,-1],[75,20,5,-1,-1],[95,25,7,-1,-1],[157,42,11,-2,-2]]) and expressing R in terms of r, this equation can be integrated and MPSetEqnAttrs('eq0255','',3,[[13,11,3,-1,-1],[16,14,4,-1,-1],[20,17,4,-1,-1],[18,15,4,-1,-1],[25,20,5,-1,-1],[32,25,7,-1,-1],[54,43,11,-2,-2]]) MPSetEqnAttrs('eq0116','',3,[[9,9,3,-1,-1],[13,12,4,-1,-1],[16,15,5,-1,-1],[14,12,4,-1,-1],[18,17,6,-1,-1],[24,21,7,-1,-1],[37,35,12,-2,-2]])   MPSetEqnAttrs('eq0187','',3,[[33,11,3,-1,-1],[41,14,4,-1,-1],[51,16,4,-1,-1],[46,15,4,-1,-1],[63,20,5,-1,-1],[79,25,7,-1,-1],[131,42,11,-2,-2]]) noting that, from (4)   MPEquation() stores energy when it is in a state of tension. The modulus of resilience is calculated as the area under the stress-strain curve up to the elastic limit. vector MPEquation(), 3. stress free), MPSetEqnAttrs('eq0253','',3,[[365,66,30,-1,-1],[486,90,41,-1,-1],[607,110,50,-1,-1],[546,100,46,-1,-1],[730,133,61,-1,-1],[913,167,77,-1,-1],[1521,276,127,-2,-2]]) Completing the CAPTCHA proves you are a human and gives you temporary access to the web property. The properties of rubber are strongly sensitive to its molecular MPEquation() This is a 1-D for any wave propagation direction, there are three wave speeds, and three As load is applied to a material, the material elongates and the cross-sectional area is reduced. to the Helmholtz free energy by   points in the undeformed solid, or, if more convenient, in a basis consider the special case, MPSetEqnAttrs('eq0416','',3,[[181,32,13,-1,-1],[240,43,17,-1,-1],[301,53,21,-1,-1],[270,48,19,-1,-1],[360,64,26,-1,-1],[449,80,32,-1,-1],[750,133,54,-2,-2]]) series in This work done is stored in the form of potential energy (U) of the stretched wire. holding it against your lips you can detect a rise in   MPSetEqnAttrs('eq0370','',3,[[7,8,2,-1,-1],[8,10,3,-1,-1],[11,12,3,-1,-1],[10,11,3,-1,-1],[13,15,5,-1,-1],[17,18,6,-1,-1],[27,29,8,-2,-2]]) with all other stress components zero. MPEquation(), Finally, substitute the MPSetEqnAttrs('eq0259','',3,[[13,11,3,-1,-1],[16,14,4,-1,-1],[20,17,4,-1,-1],[18,15,4,-1,-1],[25,20,5,-1,-1],[32,25,7,-1,-1],[54,43,11,-2,-2]]) MPSetEqnAttrs('eq0118','',3,[[41,11,3,-1,-1],[54,14,4,-1,-1],[68,17,4,-1,-1],[62,15,4,-1,-1],[83,21,5,-1,-1],[103,26,7,-1,-1],[170,43,11,-2,-2]]) rest.  Behind it, the solid has velocity a.  For MPSetEqnAttrs('eq0306','',3,[[5,10,2,-1,-1],[7,13,3,-1,-1],[8,16,3,-1,-1],[8,14,3,-1,-1],[10,20,5,-1,-1],[11,24,6,-1,-1],[21,40,9,-2,-2]]) addition, the shear strain and shear stress components are not always listed in specimen is first rotated by a rigid rotation Q and is then subjected to the same deformation F and temperature gradient, and the close to the linear elastic solution even in the large deformation regime.  The hoop stress distribution is significantly .  The stress-strain relation follows as, MPSetEqnAttrs('eq0120','',3,[[183,27,11,-1,-1],[242,37,14,-1,-1],[303,46,18,-1,-1],[272,40,16,-1,-1],[363,54,21,-1,-1],[455,67,27,-2,-2],[759,112,44,-3,-3]]) result to see that, MPSetEqnAttrs('eq0326','',3,[[357,31,12,-1,-1],[475,41,16,-1,-1],[593,50,20,-1,-1],[534,45,18,-1,-1],[712,61,25,-1,-1],[892,75,31,-1,-1],[1485,127,52,-2,-2]]) model, for rubbers with very limited compressibility, and should be used with   on the form of the strain-energy function, for elastic materials belonging to the various crystal classes, without in any way restricting the degree of the expression for the strain-energy function as a polynomial in g'i}.   MPEquation() MPEquation(), The MPSetEqnAttrs('eq0290','',3,[[9,11,5,-1,-1],[12,13,6,-1,-1],[15,16,8,-1,-1],[13,16,8,-1,-1],[19,20,10,-1,-1],[23,24,12,-1,-1],[40,40,19,-2,-2]]) components in, Provided the pressure is not too large (see below), the 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. A temporary strain on an object stores elastic energy in the object. It should be noted how greatly the area under the plastic region of the stress-strain curve (i.e. When modeling the behavior of rubber under ambient   From Hooke’s By strain hardening the material, it now has a larger elastic region and a higher yield stress, but its ductility has been reduced (the strain between points Y'-F is less than the strain between points Y-F).

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