Stress/strain responses relaxation tests

Stress relaxation tests are alternative ways of measuring the same basic phenomenon in viscoelastic polymers as creep tests, Le. the time-dependent nature of their response to an applied stress. As such, they have also been of value in understanding the behaviour of these materials. The essence of stress relaxation tests is that strain increases with time for a given stress, so that if stress is decreased with time in a controlled manner ( relaxed ), a state... 

The critical phase of either a creep test or a stress relaxation test is in the first few seconds. It is here that the transducer can be overloaded or the optical encoder can give a noisy response. Most controlled-strain instruments will give an audible error signal if the transducer is overloaded. For controlled-stress instruments, the noise level is determined by the resolution of the optical encoder. 

The viscoelastic behavior is evaluated by means of two types of methods static tests and dynamic tests. In the first calegtuy a step change of stress or strain is applied and the stress or strain response is recorded as a function of time. Stress relaxation, creep compliance, and creep recovery are static methods. The dynamic tests involve the imposition of an oscillatory strain or stress. Every technique is described in the following sections. 

Inspection of this equation shows that it models reasonably well, on a very superficial level, a stress-strain curve of the type shown in Fig. 1(b), curve (4). In other words it raises the question as to whether the deviations from linear stress-strain relationships observed in constant strain-rate tests might not be merely resulting from the intrinsic time-dependence of the linear viscoelasticity, which can be more clearly studied in creep or stress-relaxation and not due to some new process starting at high stresses. It does not take long to show that at the strain-levels of 3-5% experienced at yield, the response of most polymers is highly non-linear (r(t)/ is a function of strain-rate S as well as t, and so eqn. (14) cannot adequately describe the behaviour. However, it is also clear that at... 

In a stress relaxation test a sample is quickly placed under a strain that is then held constant, and the resulting stress is recorded as a function of time. The response of an ideal elastic solid in stress relaxation is a stress that remains constant with time, while an ideal liquid responds with an immediate return to zero stress as soon as the test strain is imposed. Viscoelastic materials respond with a stress that decays with time. Stress relaxation data are commonly reported as a time-dependent stress relaxation modulus ... 

When a sinusoidal strain is initially applied to the sample, there is a delay of at least one cycle before the stress attains the desired steady-state response. Data taken before steady state is achieved are not reliable. However, unless the frequency of the measurement is extremely low, the time required to attain steady state under a given set of conditions is much shorter than the time required to perform a creep or stress relaxation test under a single set of conditions. This short measurement time is one of the most important advantages of DMA over creep and stress relaxation. 

Indeed, carbon black-filled rubber, when loaded with time-dependent external forces, suffers a state of stress which is the superposition of two different aspects a time independent, long-term, behavior (sometimes improperly called hyperelastic ) opposed to a time dependent, short-term, response. Step-strain relaxation tests suggest that short term stresses are larger than the long term or quasi-static ones [117]. Moreover, oscillatoiy (sinusoidal) tests indicate that dissipative anelastic effects are significant, which leads to the consideration of a constitutive relation which depends not only on the current value of the strain but on the entire strain history. This assumption must be in accordance with some principles which restrict the class of rehable constitutive equations. These restrictions can be classified as physical and constitutive . The former are restrictirMis to which every rational physical theory must be subjected to, e.g., frame indifference. The latter, on the other hand, depends upon the material under consideration, e.g., internal symmetries. 

Finally, it is worth mentioning another approach used to describe nonlinear viscoelastic solids nonlinear differential viscoelasticity [49, 178, 179]. This theory has been successfully applied to model finite amplitude waves propagation [180-182]. It is the generalization to the three-dimensional nonlinear case of the rheological element composed by a dashpot in series with a spring. Thus in the simplest case, the stress depends upon the current values of strain and strain rate rally. In this sense, it can account for the nonlinear short-term response and the creep behavior, but it fails to reproduce the long-term material response (e.g., relaxation tests). The so-called Mooney-Rivlin viscoelastic material [183] and the incompressible version of the model proposed by Landau and Lifshitz [184] belraig to this class. 

The Kelvin will not have a sudden increase in strain as the damper will not allow a sudden jump in strain. Under the condition of constant stress, each model with a free damper (Maxwell and four parameter fluid) will have an ever-increasing creep strain and will be similar to the response for a thermoplastic polymer described in Fig. 3.13. Those with a free spring (Kelvin and three parameter solid) will creep to a limiting constant strain and will be similar to the response of thermoset polymers described in Fig. 3.13. In relaxation, the stress will decay to zero for those models with a free damper (Maxwell and four parameter fluid) and the stress will decay to a limiting value for those without a free damper (Kelvin and three parameter solid) as shown in Fig, 3.12 for thermoplastic and thermosetting materials respectively. Note that a simple stress relaxation test is not possible for a Kelvin model as the damper will prohibit a sudden increase in strain. 

Between the extremes of viscous fluids and elastic solids are materials that seem to exhibit both traits. These are called viscoelastic materials or memory fluids, and their dual nature becomes most evident when we subject them to time-dependent (unsteady) tests. The three major types of unsteady tests are the so-called relaxation, creep and dynamic tests. In the previous sections, we gave definitions and descriptions for stress, strain and deformation rates. These quantities are now used in defining the various unsteady tests. Thus, in a relaxation test the sample is subjected to a sudden, constant, strain. The stress shoots up in response and then gradually decays ( relaxes ). In the creep test, a sudden stress is applied and held constant. Now the strain picks up quickly and then, while continuing to increase, slows down on its rate of increase. We say the material creeps under the constant stress. In dynamic tests, one confining wall is made to move periodically with respect to another. One monitors both the strain and the stress as a function of time. 

Example 15.6 Solve Example 15.1 by applying the Boltzmann superposition principle, thereby demonstrating how the stress-time response in an engineering stress-strain test may be predicted from stress-relaxation data. 

To demonstrate that propellants are non-linear materials even at small strains, one need only check the superposition principle experimentally. In the range of small strains below detectable dewetting or volumetric dilatation [6,9-11], most propellants have a relaxation that is independent of strain and in general closely obey the scalar multiplication homogeneity rule. Yet this relaxation modulus cannot be used to accurately predict the response due to other isothermal, low rate, small strain inputs. To demonstrate the inadequacies of linear viscoelastic predictions on solid propellants, laboratory tests where superposition is applicable can be performed. Figure 6.1 illustrates the stress-strain-dilatational behavior of a typical composite propellant. The dilatation-strain behavior is caused by vacuole formation within the microstructure... 

A stress relaxation experiment can be performed on a wide range of materials. If we perform such a test on a real material a number of deviations are normally observed from the behaviour of a single Maxwell model. Some of these deviations are associated with the application of the strain itself. For example it is very difficult to apply an instantaneous strain to a sample. This influences the measured response at short experimental times. It is often difficult to apply a strain small enough to provide a linear response. A Maxwell model is only applicable to linear responses. Even if you were to imagine an experiment where a strain is... 

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