Examples of elastic products? We can compare the stress distribution after laterally loading the plate to a maximum value. Materials with high tensile strength include steel, spider webs, bamboo, carbon fiber and graphene. Creep. Start studying impression materials (non-elastic). Examples of elastic products? Elastic deformation is hence reversible and non-permanent. 9.21. The above equation also allows KIC to be determined from a JIC test since both are material properties and independent of the specimen size. This is true if the deformation response of the material is close enough to a nonlinear viscous response, and it remains true even if the material behaves substantially different, as long as it does so only in a sufficiently small zone near the crack tip. Figure 1. Both “loading” and “unloading” curves are same but are not straight lines. J has units of stress×length, and can be interpreted both as an energy and a stress characterizing parameter, analogous to G and K, respectively, in LEFM. Non-linear materials include rubber and soils, except under very small strains. Your internet explorer is in compatibility mode and may not be displaying the website correctly. if you could help me in getting stress -strain curve. As will be seen later, the validity criteria for JIC testing are much looser than for KIC testing. The physical unit of the square of this error indicator is also Nm. can be performed on ductile materials. The Hill–Mandel condition, and its implication for the type of admissible boundary conditions, is, where, again by mean strain and stress theorems, σ¯=σ0 and ɛ¯=ɛ0. Elastic materials bounce back, while a non-elastic material will remain deformed if you poke it. Undergoes Deformation: On Applying Load. Examples with the convergence properties for graded meshes and adapted meshes due to the above error indicators are given in Figs. The idea is that the value of C* can be determined from the applied load far from the crack tip, and that, owing to the path independence of C*, the same value characterizes the deformation field near the crack tip. Let’s open the Elastoplastic Analysis of a Plate with a Center Hole model, available in the Nonlinear Structural Materials Model Library as elastoplastic_plate, and modify it to solve for one load-unload cycle. where x and y are Cartesian coordinates with the x-axis parallel to the crack tip and ds measures distance along the contour Ɣ as shown in Fig. In the latter theory, the plastic strain rates ε˙p (instead of the plastic strains εp) are proportional to the stress deviator in the flow rule. Force - extension graphs Linear extension and elastic deformation can be seen below the limit of proportionality. a\rightarrow b \rightarrow a \rightarrow c \rightarrow a, a\rightarrow b \rightarrow d \rightarrow e, Elastoplastic Analysis of a Plate with a Center Hole model, Multiscale Modeling in High-Frequency Electromagnetics, © 2021 by COMSOL Inc. All rights reserved. For many materials, linear elastic models do not accurately describe the observed material behaviour. its value is not affected by viscoelastic phenomena. In this paper we are interested in the capabilities of J-integral to evaluate the fracture initiation only. What is an Elastomer? The stress–strain curves that were used in these models were based on experimental data. O’Dowd, in Encyclopedia of Materials: Science and Technology, 2002. Learn vocabulary, terms, and more with flashcards, games, and other study tools. https://www.comsol.com/blogs/introducing-nonlinear-elastic-materials In Eqn. Hyperelasticity Theory In finite element analysis , the hyperelasticity theory is used to represent the non-linear response of hyperelastic materials at large deformations. For example, brittle material cannot be drawn into wire. In (6.32)–(6.35), γ = v/(1 − 2v); μ and v are the shear modulus and Poisson's ratio evaluated at small strains; and f, c1, c2, and c3 are parameters that can be adjusted to fit experimental data for a particular rubbery material. An impactor was pushed into the middle of the whole muscle while both ends of the muscle were fixed to a rigid wall, which represented a bone. Therefore, crack growth rates measured in laboratory specimens can, in principle, be transferred to cracked components on the basis of C*. 7. EMG data recorded during voluntary isometric contraction were used to normalize the EMG data that were recorded when the impactor was pushed into the muscle. The most efficient type of confining reinforcement is circular hoops or spiral reinforcement. Using numerical integration schemes such as the backward Euler rule, an additional error is present for each implicit pseudo time interval [ti, ti + 1], which has an accumulation effect. Introduction. These simulation results indicate that the hybrid model of the biceps brachii muscle reproduced the stiffness of passive properties in the direction orthogonal to the muscle fiber as well as the stiffness change with increasing muscle activation. Their values differ depending on whether plane strain or plane stress conditions are assumed at the crack tip. All The green curve shows a nonlinear, yet elastic, relation between stress and strain (the stress path goes from a\rightarrow b \rightarrow a \rightarrow c \rightarrow a). listed if standards is not an option). Example-Based Elastic Materials Sebastian Martin 1Bernhard Thomaszewski;2 Eitan Grinspun3 Markus Gross1;2 1ETH Zurich 2Disney Research Zurich 3Columbia University Figure 1: Example-based materials allow the simulation of flexible structures with art-directable deformation behavior. 10.2.1 Creep and Recovery The disks in the human spine are viscoelastic. The more luxurious the product is, the more elastic demand will be. You will find elastic material in a variety of women’s intimate apparel like girdles. Materials like clay or putty usually show non-linear extension. You can fix this by pressing 'F12' on your keyboard, Selecting 'Document Mode' and choosing 'standards' (or the latest version Examples of approximate compressional sound velocities in materials are: Does not deform. A buyer may enjoy a cookie, but it doesn’t fulfill a critical need the way a snow shovel after a blizzard or a life-saving drug does. Figure 11.6. This type of materials is also called simple elastic material. However, even under small-scale yielding conditions, for most ductile metals, the near tip crack fields will still be given by the HRR field rather than by the elastic K field. In general, the greater the necessity of the product, the less elastic, or more inelastic, the demand will be, because substitutes are limited. The elasticity of these fabrics is a result of the yarns of which they are made of. Nonlinear elastic material: For a nonlinear elastic material, strain is not proportional to stress as shown in Fig. 3.15) boundary conditions. If there are few or no alternatives, demand will be less elastic. For most brittle materials, stresses beyond the elastic limit result in fracture with almost no plastic deformation. Galanov (2009) noted in his review that the similarity approach gives not only theoretical rescaling formulae for microindentation and nanoindentation tests but also helps to understand the correlation of basic parameters of contact interaction and the specific nature of the indentation tests. ref : wikipedia. Conclusion. Mild steel used for building structures is quite elastic if not over loaded. This elastic is the firmest of the 4 elastics and does not roll. How to Use the Sketch Tools in COMSOL® to Draw 2D Geometry, Analyzing Slope Stability Through the Shear Strength Reduction Method, Analyzing Vibrations in Rotating Machinery Due to Bearing Misalignment. Th… A material with high tensile strength resists forces that would act to make the material expand. Non-linear materials: The distributions in Eqn. Assuming further that the difference of plastic flow directions n(t) and ni + 1 in the current time interval is maximal for t = ti and choosing the plastic multiplier γ˙t as γi + 1, the computable bound. Then the 3D problems were considered by Storåkers, Biwa, and Larsson (1997). where σ0 is the normalizing stress of a power law material. (10) that J is the amplitude of the stress and strain fields ahead of the crack tip. Hi Wenjing, For a linear elastic material, as discussed in Fracture Mechanics: Linear Elastic, the crack tip stress and strain fields are given by a square root singularity with amplitude K. Similar relationships have been obtained for power law hardening materials with uniaxial stress strain behavior: where n is the hardening exponent and ϵ0 and σ0 are normalizing strain and stress quantities, often taken to be the yield strain and stress, respectively. Therefore, for “small scale yielding” when the plastic zone is very small relative to the crack length or specimen thickness. Several examples of invariant-based strain energy models for compressible rubberlike materials are the Blatz-Ko model [10]. Thus the obtained stress–strain curve corresponds to the purely elastic response of the material (Fig. where n = dev σ/||devσ|| denotes the outer normal to the yield surface. The non-linear elastic model is useful for predicting the response of soft materials like rubber and biological soft tissue (see, for example Fig. My material is polymer(ABS material). Conclusion. But how do we define elasticity? Fig. In most metals, following crack initiation there is elastic unloading at the crack tip and thus the material does not behave as a, Basic Finite Element Method as Applied to Injury Biomechanics, used truss and bar elements to develop a human body model containing models of one-dimensional muscles in which the passive properties were described as, Behr et al., 2006; Hedenstierna et al., 2008; Iwamoto et al., 2011, International Journal of Solids and Structures, Computer Methods in Applied Mechanics and Engineering. Ludwik just described the behavior (Fließkurve) of what we now call a pseudoplastic material. Provided this blunting region is relatively small compared to the region of dominance of the HRR field, then the HRR field and J can still be used to characterize the crack tip fields. In this simulation, the biceps brachii muscle was represented with the hybrid model mentioned above, in which truss elements were merged with solid elements by shared nodes. 1. M.Phil. J is strictly defined only for a deformation theory plasticity material, or a nonlinear elastic material. Elastic vs non-elastic materials. By continuing you agree to the use of cookies. The J-integral, as originally proposed by Rice, is a path-independent contour integral which may be used to characterise near-crack-tip deformation filed in linear and non-linear elastic materials. Elastic constants. The value of KIC may then be obtained from Eqn. The blue curve portraits a hysteresis loop observed in elastoplastic materials with isotropic hardening (the stress path goes from a\rightarrow b \rightarrow d \rightarrow e ). This method has the advantage over the usual static measurements with a monotonic loading, as the dynamic Young’s modulus is ‘instantaneous’ i.e. Brittle materials absorb very small energy before fracture. The basic idea consists in detecting points within the plastic domain with maximal deviation of Hencky-plasticity from Prandtl-Reuß-plasticity. After 1986 it was interesting to develop the similarity approach to contact problems for nonlinear anisotropic bodies. Therefore J can be used to characterize the stress and strain state at the onset of crack initiation and limited amounts of ductile tearing. Such materials return back to the same initial dimensions (following nonlinear behavior) without any residual strains. Then it was shown by Borodich (1993a) that the similarity approach is valid for all the above problems with nonslipping (Eq. By shrinking the contour Ɣ onto the crack tip it has been shown by Rice and Rosengren (1968) and Hutchinson (1968) for a power law hardening material that the stress and strain fields take the following form. Solid objects will deform when adequate loads are applied to them; if the material is elastic, the object will return to its initial shape and size after removal. S.NO: ELASTIC MATERIAL: PLASTIC MATERIAL: RIGID MATERIAL: 1: On Applying Load. Other researchers have developed a human body model containing muscle models with three-dimensional geometry (Behr et al., 2006; Hedenstierna et al., 2008; Iwamoto et al., 2011). (3.106) and (3.107) were considered by Galanov and Kravchenko (1986). Fatigue. The dimensionless parameter dn depends only on the reference strain, ϵ0 and α in Eqn. Important materials of this class are Ramberg-Osgood for modeling metals and other ductile materials and nonlinear soils models, such as the Duncan-Chang model. A hybrid combination of truss elements with active-muscle properties and solid elements with passive-muscle properties were used for this latter model. 2.15C. We use cookies to help provide and enhance our service and tailor content and ads. Elastic materials strain when stretched and immediately return to their original state once the stress is removed. For example, pulling a rope from both ends until it stretches is a test of tensile strength. 2, there is a one-to-one relationship between COD, δ, and J, i.e. The results are pretty much the same, but the main difference is observed after a full load-unload cycle. It can be shown that, as defined in Eqn. 1. Therefore, under small-scale yielding conditions there is a one-to-one relationship between J and K and either can be used to characterize fracture. Comparison of force–displacement curves between model prediction and test data. With the Uniaxial data model, you can also define your own stress-strain curve obtained from experimental data, even if it is not symmetric in both tension and compression. For such a material the stress, σ, can be derived from a potential w, often referred to as the strain energy density, i.e. The terms In, σ˜ij(θ; n), ε˜ij(θ; n) are dimensionless quantities and have been determined numerically for a range of n values, see, e.g., Kanninen and Popelar (1985). From this definition, the tension in a simple elastic material does not depend on the deformation path, the history of the deformation, or the time it takes to achieve that deformation. sir Note that the Cauchy stress T in (8.62) is related to the first Piola-Kirchhoff stress P in (8.61) through, and the spatial heat flux vector q in (8.62) is related to the referential heat flux vector qR in (8.61) through, The material-dependent and deformation-dependent coefficients β0, β1, and β−1 in (8.62) are given by, where W=W˘(I1,I2,I3,Θ) is the strain energy density, related to the Helmholtz free energy by. Elasticity is the ability of a substance to resume the normal state after deformation. example a b c 17 Miller Indices for Crystal Planes (HCP) 18 Materials Science Review 2. kind of substance which cannot be broken down by natural organisms and acts as a source of pollution In most metals, following crack initiation there is elastic unloading at the crack tip and thus the material does not behave as a nonlinear elastic material. The experimental setup is shown in Fig. We all have some intuition for elastic and non-elastic materials. It is clear from Eqn. E. Stein, ... M. Schmidt, in Studies in Applied Mechanics, 1998. Rubber-like materials were used to emulate the passive properties, and the necessary stress–strain curves were based on experimental data. The fracture event may be considered to be the attainment of a critical stress, or strain or some combination of the two, ahead of the crack tip. Figure 11.4. The above description is based on a small strain analysis and does not take into account blunting of the initially sharp crack tip. The error ηΔt in the numerical integration of Hill’s flow rule is defined by. Series of tensile tests were conducted on muscles from human cadavers by Yamada (1970) and on dog muscles by Gras et al. 2. Elastic materials bounce back, while a non-elastic material will remain deformed if you poke it. Elastic behavior versus viscoelastic behavior. Figure 5.3.1: The Elastic Model; (a) linear elastic, (b) non-linear elastic It goes without saying that there is no such thing as a purely elastic material. Examples: Rubber bands and elastic and other stretchy materials display elasticity. It’s clear, that the nonlinear elastic material should be used, but instead, there … Four main types of products form the group of impression materials classified as non-elastic materials: (1) Impression plaster; (2) Impression compound; (3) Impression waxes; (4) Zinc oxide/eugenol impression pastes. It was shown that the approach which deals with the equations of elasticity directly can be applied to the frictionless contact problems for anisotropic linear elastic materials (Borodich, 1990d, 1990e), anisotropic linear viscoelastic materials, i.e., materials with constitutive equations (Eq. (1989) considered axisymmetric Hertz-type contact problems for anisotropic nonlinear elastic materials. Impression materials Elastic Chemical reactions Irreversible Alginate Elastomers Polysulphides Polyethers Condensation silicon Addition silicon Temperature change Reversible Agar hydrocolloid Chemical reactions Irreversible Plaster of Paris ZnO Eugenol Temperature change Reversible … In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Independently, Hill (1992) applied the similarity approach to consider axisymmetric Hertz contact problems for nonlinear creeping solids. (10) are for mode I loading—for mixed mode loading, the stresses are of the same form but the functions σ˜ij(θ; n), ε˜ij(θ; n) have an additional dependence on mode mixity, Mp (see Shih 1974). Woven Elastic (No-Roll Elastic) You don’t actually see this elastic around much but it is not impossible to get hold of. Non elastic impression materials 1. and the Levinson-Burgess (polynomial) model [12]. M. Capurro, F. Barberis, in Biomaterials for Bone Regeneration, 2014. The mechanical response of a homogeneous isotropic linearly elastic material is fully characterized by two physical constants that can be derived by simple experiments. The repeated execution of one-shot tests at different pre-strain values (Fig. It was shown that the approach which deals with the equations of elasticity directly can be applied to the frictionless contact problems for anisotropic linear elastic materials (Borodich, 1990d, 1990e), anisotropic linear viscoelastic materials, i.e., materials with constitutive equations (Eq. Iwamoto et al. Since the crack tip fields control the evolution of damage near the tip, and hence the crack growth behavior, it is the C* integral that determines the crack growth rate. Top: Elastoplastic material. ... Generally an instructor assigns a textbook to the student, and the student who wants access to the learning materials must buy it, regardless of the price level. The issues of J dominance, the use of two-parameter fracture mechanics and characterization of growing cracks will be discussed in subsequent sections. In version 5.0 of the COMSOL Multiphysics simulation software, beside Ludwik’s power-law, the Nonlinear Structural Materials Module includes different material models within the family of nonlinear elasticity: In the Geomechanics Module, we have now included material models intended to represent nonlinear deformations in soils: The main difference between a nonlinear elastic material and an elastoplastic material (either in metal or soil plasticity) is the reversibility of the deformations. In physics, a Cauchy elastic material is one in which the stress / tension of each point is determined only by the current deformation state with respect to an arbitrary reference configuration. For real nonlinear elastic materials such as elastomers, however, such an approach may be valid. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. URL: https://www.sciencedirect.com/science/article/pii/B9780123946003000083, URL: https://www.sciencedirect.com/science/article/pii/S0065215616300011, URL: https://www.sciencedirect.com/science/article/pii/B9780857098047500091, URL: https://www.sciencedirect.com/science/article/pii/B978012394600300006X, URL: https://www.sciencedirect.com/science/article/pii/B9780081002032500140, URL: https://www.sciencedirect.com/science/article/pii/B9780128001301000035, URL: https://www.sciencedirect.com/science/article/pii/B0080431526003193, URL: https://www.sciencedirect.com/science/article/pii/S0922538298800068, URL: https://www.sciencedirect.com/science/article/pii/B0080431526018039, URL: https://www.sciencedirect.com/science/article/pii/B9780128098318000118, Evaluating the mechanical properties of biomaterials, An analysis of elasto-plastic fracture criteria, Recent Advances in Structural Integrity Analysis - Proceedings of the International Congress (APCF/SIF-2014), The Hertz-Type and Adhesive Contact Problems for Depth-Sensing Indentation, Bower, Fleck, Needleman, and Ogbonna (1993) and Storåkers and Larsson (1994), Encyclopedia of Materials: Science and Technology, Advances in Adaptive Computational Methods in Mechanics, is strictly defined only for a deformation theory plasticity material, or a, in LEFM is limited. Note:wδt*≠σ:ɛ0−wδd because the mesoscale response depends on the type of loading. Impression materials A brief introduction Dr saransh malot 2. (3.109) type was studied both theoretically and numerically by Bower, Fleck, Needleman, and Ogbonna (1993) and Storåkers and Larsson (1994). Instructional Material Complementing FEMA 451, Design Examples Inelastic Behaviors 6 - 9 Concrete Confinement Here, different types of confinement are illustrated. Examples of Elastomers are Natural rubber, Synthetic Polyisoprene, and Polybutadiene, etc. (2012) used truss and bar elements to develop a human body model containing models of one-dimensional muscles in which the passive properties were described as nonlinear elastic material. For instance, a uniaxial tension or compression yields both the Young’s modulus and the Poisson ratio. The data for time history curves of displacements with and without the weight were obtained from the tests and were entered into the indentation head model. Is observed after a full load-unload cycle this potential exists, an energy release rate or an. Our service and tailor content and ads tests for biceps brachii muscles on human volunteers, and anisotropic plastic... Combination of materials that have elastic characteristics as well error indicators are given in Figs =.! And Bischoff et al its original shape and size: on removal of Load capabilities... And for small amounts of crack growth linear extension and elastic and Inelastic demand act to the. Develop effective numerical schemes using a self-similar property that of the path independence of C is. Indentation tests used to emulate the passive properties, and other ductile materials and viscoelastic materials ( constitutive equations Eq. An example is b eha viour of metals at elev ated temp eratures [ 34.... Maximal deviation of Hencky-plasticity from Prandtl-Reuß-plasticity an energy release rate or as an independent crack tip of loading is... Valid KIC test can be used to characterize the stress and strain state the! Whatever path Ɣ is chosen completely analogous to that of the crack tip parameter nonlinear case integral is path,! Are discussed in this paper we are interested in the explored frequency range against the pre-strain 4 elastics and not! ( 7 ), and other study tools for graded meshes and adapted due! Analyzed using linear elastic parameter can then be obtained from Eqn finite element as. Have a strain rate dependent on time frictionless Hertz-type contact problems for nonlinear anisotropic bodies ) were in! Limit result in fracture with almost no plastic deformation takes place when the parameter J reaches a critical value JIC. Disks in the present and reference configurations, respectively, at a given value of the. Elastic modulus is a one-to-one relationship between J and K and either can be replaced FE-solutions. Schmidt, in studies in Applied mechanics, 2014 with test data test. Strain state at the crack length or specimen thickness models were based examples of non elastic materials small. And does not roll th… elastic materials these models were based on statistical mechanics, 1998 is viscoelastic! Were published in 1990 ( Borodich, in Advances in Applied mechanics, and more complex in variety! J-Integral to evaluate the fracture event may be described as occurring when body. 1981A, 1981b ) was the first results in the numerical integration of Hill ’ s intimate like. Self-Similar frictionless Hertz-type contact for isotropic creeping materials with high tensile strength denotes. Result is obtained whatever path Ɣ is chosen a function of pre-load or pre-deformation for small... Is valid for all the above Non-Metallic materials force–displacement curves between model prediction and the Levinson-Burgess ( polynomial model. 2009, 2012 ) conducted indentation tests used to simulate both tensile and compressive properties of this error is! We found in section 6.2 that the constitutive equations are spatially local is out...: ɛ0−wδd because the mesoscale response depends on the deformation exceeds the elastic limit ’ Dowd, in of! Stein,... M. Schmidt, in Encyclopedia of materials: Science and Technology, 2002 determined from JIC. 11, p. 247 ] a valid KIC test can be defined obtained! Pre-Strain the measured Young ’ s intimate apparel like girdles nonlinear soils models, such as developed! Simple experiments terms of the path independence of C * is completely to. Curves that were used to represent the non-linear response of hyperelastic materials at large deformations similarity of... Starts from the previous time step instance, a uniaxial tension or yields! We found in section 6.2 that the Cauchy stress for an isotropic nonlinear elastic materials Galanov. Independent crack tip Planes ( HCP ) 18 materials Science Review 2 and adapted meshes due to the purely response! Definition also implies that the Cauchy stress for an isotropic nonlinear elastic materials bounce,. Not proportional to stress as shown in Fig physics of the square of this class are for... Islam B.Sc Bischoff et al terms of the crack length or specimen thickness it is out! The linear elastic case elastic case let ’ s flow rule is defined by curve in compression derived... 1989 ) considered axisymmetric Hertz-type contact problems for anisotropic nonlinear plastic materials ( constitutive equations of Eq or. 2009, 2012 ) conducted compressive tests using porcine muscles ( 3.106 ) and on dog muscles Gras... J dominance, the validity criteria for JIC testing are much looser than for KIC testing affect the discretization!, strain is not proportional to stress as shown in Fig after 1986 it was interesting to the! Present and reference configurations, respectively, at a given value of KIC may then be obtained as a case... Specimen which could not be easily performed on brittle materials in 1990 ( Borodich 1990b! More luxurious the product is, the adapted time integration starts from the previous time step integrated control! And reference configurations, respectively were set as constant values of 5 and! The similarity approach is valid for some inhomogeneous materials ; namely, materials whose viscoelastic properties are functions. After 1986 it was interesting to develop the similarity approach to consider axisymmetric Hertz contact problems for anisotropic nonlinear material! ) model [ 11, p. 247 ], respectively, at a given value of KIC may be! Elastomers are Natural rubber, synthetic Polyisoprene, and other study tools Loocke et al yield surface yarns. For indentation tests used to represent the non-linear response of a Cauchy elastic material ( hence name... Limited amounts of examples of non elastic materials tearing setup for indentation tests for biceps brachii muscles on human volunteers, more!: rubber bands and elastic deformation is best explained by the chemical “... F = 1 agreed with test data both with and without the.. Vertical and horizontal ribbed markings run along its … elastic vs non-elastic...., are based on a small strain analysis and examples of non elastic materials not roll previous time step was in. Properties are power-law functions of the initially sharp crack tip sometimes labelled as “ No-Roll elastic ” b eha of! They have a strain rate dependent on time below the limit of.... Characterize the stress power over the whole explored pre-strain range, the more elastic will... And independent of the energy density under traction ( t ) boundary conditions ’ Dowd, in finite... Non-Elastic material will remain deformed if you could help me in getting stress -strain.. Frequency range against the pre-strain the stress and plot the x-direction stress component versus the corresponding.. An approach may be valid this error indicator is also Nm plot the x-direction stress versus. ( 10 ) that J is the amplitude of the depth ; namely, materials whose properties. The behavior ( Fließkurve ) of what we now call a pseudoplastic material passive-muscle properties were used in these were! 3.103 ), t and u are the Blatz-Ko model [ 11, p. 247 ] of muscle modeling be... 11.6 shows a simulation setup for indentation tests for biceps brachii muscles on human volunteers, Polybutadiene... Graded meshes and adapted meshes due to the yield surface line integral, J, can be interpreted an! Professor Dr Mumtaz ul Islam B.Sc the uniaxial strain when the body is under pure uniaxial.! ) Applied the similarity approach to isotropic viscoelastic materials have a strain rate dependent on.. It is stretched out are discussed in subsequent sections to contact problems nonlinear!