Caroline Lancelot Miltgen

Gaëlle Pantin

Bianca Grohmann

Communicating Sensory Attributes and Innovation Through Food Product Labeling

Keywords: Communication of sensory information, consumer expectations, new food products, packaging, sensory attributes * 0.1

Purpose -This research explores the influence of informational stimuli on product sensory expectations and perceived newness of the product.

Design/methodology/approach -Two experiments examine to what extent consumers use product typicality, graphical representations, and package typicality in evaluating new products.

Findings -Study 1 finds that (1) a typical flavor induces more positive expectations of pleasantness, taste, color, and smell, and (2) the presence of graphic representation on product labels increases perceived pleasantness, but does not affect sensory expectations. Study 2 indicates that the product seems newer in the absence of a package (label-only condition) but when the product packaging is presented, an atypical package conveys more newness than a typical package.

Originality/value -These results provide practical guidelines for the design and introduction of innovative food products, which can help practitioners use appropriate communication cues to evoke positive sensory expectations and anticipated pleasantness.

Introduction

At the point of purchase, informational stimuli such as product labels and packaging play a fundamental role in consumers' purchasing decisions, especially for new food products.

Consumers learn about new products in many ways [START_REF] Hoeffler | Measuring preferences for really new products[END_REF], including categorybased learning, analogies, and mental simulations. Understanding how informational stimuli affect consumer perceptions, evaluations, and behaviors has both theoretical and managerial implications [START_REF] Imram | The role of visual cues in consumer perception and acceptance of food product[END_REF]. This research builds on cognitive psychology research pertaining to the role of typicality and visual information to explore the role of labels and packaging as antecedents of taste and pleasantness perceptions for a new food product. It examines the influence of informational stimuli (i.e., labels, packaging) on expectations about a new food product (flavored cider) and focuses on the influence of the typicality of the flavor associated with the product, the type of packaging, and the presence/absence of graphical representations on the label on consumers' sensory expectations, evaluations of pleasantness, and perceived newness. This research contributes to theory and managerial practice by identifying product characteristics that convey the newness of the product and by testing to what extent images and colors and packaging-which are often the first cues consumers are exposed to-determine consumers' new product-related beliefs.

Antecedents of Expected Sensory Perception and Pleasantness

Product typicality

When consumers consider a food product, its sensory properties-appearance (sight), smell (olfaction), texture (vision, touch, proprioception), and to a lesser extent sound (hearing)influence evaluations [START_REF] Delwiche | The impact of perceptual interactions on perceived flavor[END_REF]. The information provided by sensory properties help consumers categorize the product, form expectations about it, and anticipate pleasure based on previous experiences. For food products, taste is an important evaluation criterion. To anticipate a new product's taste, consumers use past experiences-if available-or form inferences based on the product's extrinsic features [START_REF] Pinson | An implicit product theory approach to consumers inferential judgments about products[END_REF]. In the case of new products, consumers initially have to rely on extrinsic cues such as visual information on labels and packaging to evaluate the product. Based on extrinsic cues, consumers develop sensory expectations and expectations about the potential pleasure of ingesting the product.

Thus, extrinsic cues (e.g., price, brand, labels, packaging) rather than the product itself initially influence consumers' beliefs about a product. Extrinsic cues generate cognitive reactions-such as sensory expectations (e.g., about product taste)-or emotional responses highlighted by hedonic expectations (e.g., pleasure expected from consuming a food product).

For example, color influences predictions about the intensity of the taste and smell of beverages [START_REF] Franck | Strawberry odor, but not red color, enhances the sweetness of sucrose solutions[END_REF][START_REF] Pangborn | Influence of color on the discrimination of sweetness[END_REF][START_REF] Pangborn | The influence of color on discrimination of sweetness and sourness in pear nectar[END_REF][START_REF] Zellner | Effects of color on expected and experienced Refreshment, Intensity, and liking of beverages[END_REF][START_REF] Zellner | Color affects perceived odor intensity[END_REF]. For food, the flavor associated with the product also contributes information, such that familiarity with the flavor and its congruence with the specific food item influence evaluations [START_REF] Porcherot | Contribution à la mesure de la familiarité et de la complexité d'arômes alimentaires. Pertinence de ces concepts pour expliquer les préférences des consommateurs[END_REF]. Flavor familiarity thus has a significant effect on the acceptability of food products (Stalleberg-White and Pliner, 1999), especially for innovations. Aromatic (i.e., smell) complexity also may affect palatability, according to people's experience and the novelty of the stimulus, but this component has not been studied as much due to its methodological challenges [START_REF] Levy | Perceived stimulus complexity and food preference development[END_REF][START_REF] Sulmont-Rossé | Impact of the arousal potential of uncommon drinks on the repeated exposure effect[END_REF]. This research considers the influence of the product's flavor typicality on consumers' expectations about the odor, color, and taste of a new product, as well as the degree of anticipated pleasantness of the product. Typicality describes the degree to which a product is representative of a category, and is based on prior learning experiences and associations [START_REF] Heckler | The role of expectancy and relevancy in memory for verbal and visual information: what is incongruency?[END_REF]. Atypical flavors are more difficult to categorize, and should therefore lead to more negative sensory and pleasantness expectations. H1: For a new food product, information regarding a typical (vs. atypical) flavor prompts more positive expectations of (a) pleasantness, (b) taste, (c) color, and (d) smell.

Product information

Product information should be presented in a manner that increases the ease of comprehension, such as by using "symbols which quickly convey the concept" [START_REF] Bettman | Consumer psychology[END_REF]p.15). Dual processing theory suggests that when an icon reinforces a verbal statement, the two components combine to improve overall understanding and recall. [START_REF] Heckler | The role of expectancy and relevancy in memory for verbal and visual information: what is incongruency?[END_REF], for example, find that memory improves when a message includes a congruent (i.e., both relevant and expected in the context) image. The influence of descriptions on product labels has been investigated, but never explicitly for innovative products. For example, diet and light labels significantly influence sales and sensory evaluations of food products [START_REF] Wansink | How diet and health labels influence taste and satiation[END_REF], though the influence varies according to the hedonic (e.g. a dessert) or functional (e.g. an entree) nature of the food product. [START_REF] Wansink | How descriptive food names bias sensory perceptions in restaurants[END_REF] also demonstrate that a restaurant menu that offers evocative, descriptive suggestions (e.g., "delicious fish filet, savory Italian style") leads consumers to consider the food as more appetizing, tasty, and calorie-laden, compared to the use of simple terms (e.g., "fish filet"). These effects are not limited to sensory descriptions: Keller and colleagues (1997) show that nutrient levels listed in Nutrition Facts panels on the back of packages mitigate the effect of nutrition claims on the front of the packages on consumers' overall product evaluations and purchase intentions. This suggests that consumers do not rely primarily on nutrition claims to make their overall nutrition and product evaluations, if other nutritional information is readily available [START_REF] Ford | Can consumers interpret nutrition information in the presence of a health claim? A laboratory investigation[END_REF][START_REF] Keller | The effects of nutrition package claims, nutrition facts panels, and motivation to p rocess nutrition i nformation on c onsumer p roduct e valuations[END_REF].

In marketing communications and packaging, visual elements often dominate verbal elements in their effect on information processing, due to their vividness and ability to attract attention [START_REF] Mitchell | The effects of verbal versus visual components of advertisements on brand attitudes and attitude toward the advertisement[END_REF][START_REF] Paivio | Picture superiority in free recall imagery or dual coding?[END_REF]. Images are easy to recognize and remember [START_REF] Childers | Conditions for a picture-superiority effects in consumer memory[END_REF], and have a profound impact on consumer perceptions and judgment [START_REF] Holbrook | Feature interactions in consumer judgments of verbal versus pictured presentations[END_REF]. Graphical representations exert significant, long-term effects on product beliefs and purchase intentions [START_REF] Bone | Package graphics and consumer product beliefs[END_REF]. The current research tests whether graphical representations (presence vs. absence of ingredient images) influence consumers' sensory and pleasantness expectations about a new food product.

Effects of Package Typicality on Sensory Expectations and Perceived Newness

Package typicality, sensory and pleasantness expectations

Research acknowledges the importance of package design [START_REF] Bloch | Seeking the ideal form: product design and consumer response[END_REF][START_REF] Garber | The role of package color in consumer purchase consideration and choice[END_REF][START_REF] Hertenstein | The impact of design effectiveness on corporate financial performance[END_REF][START_REF] Rettie | The verbal and visual c omponents of package design[END_REF][START_REF] Roehm | The relationship between packaging uniformity and variety seeking[END_REF][START_REF] Schoormans | The effect of new package on product attention, categorization and evaluation[END_REF][START_REF] Veryzer | A nonconscious processing explanation of consumer response to product design[END_REF], because package design distinguishes brand offerings and induces favorable impressions [START_REF] Berkowitz | Product shape as a design innovation strategy[END_REF][START_REF] Henderson | Building strong b rands in Asia: Selecting the v isual c omponents of i mage to m aximize brand strength[END_REF][START_REF] Schmitt | Marketing aesthetics: the strategic management of branding, identity and image[END_REF]. Decisions about package typicality are critical for managers, and include considerations of brand image associations (e.g., Coca-Cola bottle) and consumer perceptions.

For example, packaging that appears larger tends to be purchased more often (Raghubir and Krischna, 1999). During their in-store shopping, consumers exert little effort to find volume information on package labels [START_REF] Cole | Age differences in consumers' search for information: Public policy implications[END_REF]. Furthermore, packages which attract more attention are also perceived to contain more volume of a product than similarly sized packages that attract less attention. The disparity in the level of attention creates a mental "contamination" of volume assessment [START_REF] Folkes | The e ffect of package shape on consumers' juudgments of product volume: Attention as a mental contaminant[END_REF]. Thus, packaging enables managers to alter consumers' product perceptions without substantively changing the product [START_REF] Bloch | Seeking the ideal form: product design and consumer response[END_REF][START_REF] Wansink | Can package size accelerate usage volume?[END_REF].

Various marketing communication cues (labels and package typicality for example in terms of shape) may be influential and include information about product typicality and graphical representations on labels. Marketers should convey consistent messages across different elements of the communication mix, whether in advertisements, in-store promotions, or packaging [START_REF] Shimp | Advertising, Promotion, and Other Aspects of Integrated Marketing Communications[END_REF]. Cue consistency has important implications for brand equity and sales [START_REF] Duncan | Client perceptions of integrated marketing communications[END_REF][START_REF] Low | Correlates of integrated marketing communications[END_REF][START_REF] Mcarthur | A marketing management view of integrated marketing communications[END_REF]. [START_REF] Garretson | The role of spokescharacters as advertisement and package cues in integrated marketing communications[END_REF] examine strategic combinations of advertising cues, including spokespersons and verbal attributes in integrated marketing communications (IMC) campaigns and find empirical evidence of the benefits of including consistent spokes characters throughout IMC campaigns, including favorable attitudes toward a brand. The current research examines package typicality and its impact on consumers' perceptions of sensory attributes and pleasantness. In light of the evidence regarding the positive impact of cue consistency, it includes package typicality (label only, bottle, can) along with ingredient typicality (caramel vs. black currant) and ingredient information (absence vs. presence of graphical representation) as experimental factors to explore potential interaction effects. In line with the literature on typicality, the following hypotheses arise: H3: A typical package increases expectations of (a) pleasantness, (b) taste, (c) color, and (d) smell compared to an atypical package, and even more so compared to a label only.

Package typicality and perceived newness of the product

Consumers are generally ambivalent toward product innovativeness, especially in the food sector, and may wish to both approach (neophilia) and avoid (neophobia) new food products.

The opposing tendencies driven by curiosity and fear result in consumer preferences for new products with moderate levels of innovation [START_REF] Spielberger | Curiosity and anxiety[END_REF][START_REF] Zuckerman | Psychobiology of Personality[END_REF]. [START_REF] Folkes | The e ffect of package shape on consumers' juudgments of product volume: Attention as a mental contaminant[END_REF] suggest that a package that attracts more consumer attention (less typical) appears newer than a similarly sized package than attracts less attention (more typical). The current research considers cider as the focal product category. As cider is usually packaged in a bottle when sold in supermarkets, local street markets and wine merchants, the bottle is considered as the more typical package. A can (i.e., atypical package) should thus attract more attention than a bottle, because of its novelty [START_REF] Johnson | Attention capture by novel stimuli[END_REF].

Regardless of the packaging, an atypical flavor should also attract more attention. Thus perceived newness should be higher when the consumer sees only the label and not the product package. If consumers judge a product based on only its label, information on the label receives undivided attention (i.e., is not directed toward package typicality) and thus highlights ingredients and flavors. However, product packaging should reinforce product newness if that packaging itself is atypical, or diminish perceived product newness if packaging is typical. H4: The perceived newness of a new food product is higher when consumers only see the label rather than the product packaging (H4a). Perceived newness is higher when the product is contained in an atypical packaging than when it is contained in a typical package (H4b). [START_REF] Radford | Liking innovation to design: consumer responses to visual product newness[END_REF] underlined that the connection between product innovation and visual design remains understudied despite being recognized as an important issue by innovation researchers [START_REF] Hauser | Research on innovation: A review and agenda for Marketing Science[END_REF] and the growing interest in consumer reactions to design generally (e.g. [START_REF] Bloch | Seeking the ideal form: product design and consumer response[END_REF][START_REF] Leder | A model of aesthetic appreciation and aesthetic judgments[END_REF][START_REF] Veryzer | The influence of unity and prototypicality on aesthetic responses to new product designs[END_REF]. Study 2 examines the effect of packaging typicality on sensory expectations and perceived newness (H3 and H4) in order to bring new insight to the product innovation and visual design literatures.

Study 1

This experiment investigates how the typicality in flavors and graphical information influence product evaluations and expected liking in the absence of an actual taste experience.

Stimuli and procedure

Stimuli consisted of labels for an alcoholic cider beverage with a fictitious brand name.

Research using fictitious brands contributes more research knowledge [START_REF] Campbell | Brand familiarity and advertising repetition effects[END_REF] because fictitious brands serve as novel brand, and findings therefore extend to prototype development scenarios. Product typicality was manipulated through product flavor: black currant (typical) and caramel (atypical). The ingredients were clearly specified on the labels by flavor name and associated colors (purple, yellow-orange; see Appendix A).

Furthermore, graphical representation of ingredients (bunch of black currants, caramel chunks) was present or absent. The experiment thus employed a 2 (ingredient typicality: typical, atypical)  2 (graphic representation: absent, present) between-participants design.

In an online survey, 460 French university students (20.5 years of age on average 1 ) were contacted via e-mails that contained a link to the online questionnaire. They were asked to evaluate a new beverage in advance of its market launch. Cider was selected because it is a popular alternative drink for younger consumers in search of novelty and contains less alcohol 1 Although the legal drinking age is 21 years in the United States, it is only 18 in the country where the study was run. The students surveyed would thus legally be able to have prior experiences with alcoholic ciders and to give their anticipated reactions to a new alcoholic beverage such as the one proposed here.

than other drink mixes. It also requires new consumption and purchasing habits without involving a new production method.

Participants were randomly assigned to one of the conditions and first saw an image of the four experimental stimuli (product labels). Participants then answered five-point Likert-scales (anchored fully disagree/fully agree) measuring sensory expectations (taste, smell, color; two items each) and pleasantness (two items). Three items measured the perceived typicality of the flavor ("This aroma accompanies the cider well", "It seems to be a good mixture", "The flavor matches the product category") and served as manipulation checks.

Results

Table 1 summarizes the MANOVA results.

Insert Table 1 about here

In support of H1, typical ingredients (i.e., black currant) induce more positive product expectations in terms of pleasantness, taste, color and smell. Specifically, the typical product was perceived as more tangy (3.45 vs. 3.04, p = .000) and fruitier (4.02 vs. 2.92, p = .000), with an original color (3.57 vs. 3.09, p = .000) and a pleasant smell (3.04 vs. 2.85, p = .032), and it was expected to be more agreeable to drink (3.04 vs. 2.63, p = .000). In contrast, the atypical product appeared more disagreeable (3.04 vs. 2.70, p = .000) and sickening (3.54 vs.

2.97, p = .000).

The presence of a visual representation of the flavor on the label induced more positive expectations about product pleasantness. The typical product was expected to be more pleasant to drink (2.92 vs. 2.72, p = .041) and less sickening (3.16 vs. 3.40, p = .033), in support of H2a. However, the presence or absence of a visual representation did not affect disagreeable taste (2.96 vs. 2.81, p = .146), different smell (2.90 vs. 2.98, p = .413), and color originality (3.31 vs. 3.34, p = .803) expectations. Visual representation thus did not influence taste, color, and smell expectations. H2b-H2d were not supported.

Finally, a significant interaction effect emerged between flavor typicality and the absence/presence of graphical representation of the flavor with regard to taste expectations: the typical product and presence of the ingredient image on the label resulted in fruitier product expectations (4.16), compared to absence of ingredient image (3.81, p = 0.011). This experiment supports the predicted effect of product typicality on consumers' sensory expectations. As hypothesized, consumers appear to prefer a typical versus atypical/moderately atypical flavor, confirming a preference for the norm [START_REF] Campbell | The moderating effect of perceived risk on consumer's evaluation of product incongruity: preference for the norm[END_REF]. In addition, the presence of a visual representation on the label induced more positive expectations about the pleasantness of the product confirming that graphical representations exert a significant effect on consumers' product beliefs [START_REF] Bone | Package graphics and consumer product beliefs[END_REF]. The difference between this study and earlier work is that it considers the joint effects of product typicality and presence/absence of a visual representation on taste expectations and pleasantness.

Study 2

Study 1 focused on the impact of informational stimuli (information regarding product typicality and visual representation) on consumer expectations. To examine the effect of package typicality on expectations and product newness perceptions (H3 and H4), a second study was conducted. Cider again served as the focal product category.

Stimuli and procedure

Study 2 uses a 2 (ingredient typicality: typical, atypical)  2 (graphic representation: present, absent)  3 (package typicality: label-only, typical [bottle] versus atypical [can] package) between-participants design. To control for possible confounds, both packages were depicted in the same material (metal). A three-item manipulation check for product typicality confirmed that black currant cider more typical than caramel flavored cider (3.40 vs. 2.91; F = 29,742; p = .001).

The questionnaire was completed by 383 students and consumers under 35 years of age (92.4% younger than 24 years, 6.3% between 25-34 years; 98.4% students; 55.5% women), who represent the main target market for alcoholic ciders. After being randomly assigned to one of the experimental conditions, participants rated sensory expectations (taste, smell, color) on two items each, and expected pleasantness on four items. Perceived newness was measured on two items ("original," "inspired"). Finally, participants reported their usual cider consumption habits.

Results

An exploratory factor analysis for the pleasantness scale confirmed a single dimension and satisfactory reliability (α = .79). The results shown in Tables 2 and3 summarize the results of this study.

Insert Tables 2 and3 about here The typical packaging increased perceived pleasantness compared to label-only and even more so compared to the atypical packaging (3.30 vs. 3.16 vs. 2.75, p = .000), supporting H3a. However, H3b was not supported in terms of both fruity taste (p = .499) and disagreeable taste (p = .152). The effect of the package on color perceptions was only partially supported (H3c) in that respondents perceived color as more original in the label-only and atypical packaging conditions (3.57 vs. 3.48 vs. 3.33, p = .024). Color was perceived as more distant for the atypical packaging, compared to label-only and typical packaging. Similarly, the effect of packaging on smell perceptions was only partially supported (H3d): pleasant smell expectations were higher for the typical packaging compared to the label-only or atypical packaging (3.40 vs. 3.25 vs. 2.96, p = .000). In contrast, the atypical packaging resulted in higher scores on "special smell" compared to the label-only or typical packaging (p = .450).

Newness was perceived as higher in the label-only version, which supports H4a. The atypical packaging generated more perceived newness on the item "imaginative", but the typical package was rated higher on the item "original". H4b was therefore partially supported.

This study finds several interaction effects of package typicality and graphical representation on color expectation ("original color"), product originality, and pleasantness about the product (pleasantness; see Appendix B): Color expectation differed across graphical representation conditions in the label-only condition, but not in the atypical packaging condition. For the typical package, color was expected to be more original in absence of the ingredient whereas the contrary holds for the label-only condition.

The results for the product global originality item suggest that the typical package with a graphical representation of ingredients was expected to be less original. The product appeared more original for label-only combined with graphical representation of ingredients or for the typical package without graphical representation. Finally, in terms of pleasantness evaluations, the typical packaging without graphic representation scored highest, whereas the atypical package without graphic representation scored lowest.

This research focuses on (visual) informational stimuli as determinants of sensory expectations, pleasantness, and newness perceptions. The interaction effects shed more light on how consumers react to an atypical (new) product. For the typical package, color and product are expected to be more original in the absence of graphical representation of ingredients, which means that graphical representation is not useful in conveying originality.

For managers, it is important to consider the need for and impact of graphical representation on sensory expectations and perception of newness.

General Discussion

Implications

This research demonstrates that design elements can be used strategically to create favorable expectation of food products in the absence of taste experience. People infer taste, color, and quality of a new product on the basis of extrinsic cues which are under marketers' control. For example, the combination of a fruit-derived beverage (cider) with another, typical fruit ingredient (black currant) generated more positive expectations than a combination with an atypical ingredient (caramel). Respondents predicted that caramel cider would be sweeter, more sickening, and disagreeable in its aftertaste, whereas black currant cider was expected to be pleasant to drink. Sensory product evaluations depend significantly on ingredient combinations, and this research supports that typicality significantly influences consumers' evaluations, including their taste perceptions and anticipation of pleasantness.

Furthermore, graphic representation provided on product labels influences consumers' taste and pleasantness predictions. Graphic representation of ingredients made the new product more pleasant and less sickening, maybe because it diminish perceived risk. The current empirical validation of the relation between informational cues about the product and sensory perceptions provides an important theoretical contribution regarding sensory interactions in the food sector.

Theoretical contributions

This research differs from previous investigations in that it examines several informational cues (product typicality, graphic representation, package typicality) that affect various dimensions of sensory expectations. Package typicality, for example, affects product evaluation, including expected pleasantness. Packaging further affects the perceived sweetness or acidity of its contents, which indicates that consumers infer gustatory beliefs from extrinsic cues. This research thus extends the literature on the impact of packaging variables on consumer reactions. Whereas previous studies focused mainly on the shape or color of products, sensory marketing demands more complex approaches that combine a greater number of variables. Manipulation of various cues therefore increases current knowledge of sensory interactions among product, labels, and packaging. This research complement work by [START_REF] Rettie | The verbal and visual c omponents of package design[END_REF] regarding the quantity and location of information contained on packages. The presence of graphical representations as indicators of flavor influenced taste expectations; the presence/absence of graphical representations affect expected taste particularly strongly when ingredients are typical of the product. For example, an image of black currants made the cider seem less sweet-perhaps because it conjured the astringent taste of black currents to a greater degree. However, an image of caramel chunks did not influence sweetness intensity ratings.

This research speaks to the potential impact of the location of the product image, such that images on the front label may influence consumers' perceptions and evaluations to a greater or lesser extent depending on their location [START_REF] Deng | Is your product on the right side? The location effect on perceived product heaviness and package evaluation[END_REF]. Further research can pursue effects of the location of graphical ingredient representations beyond the current focus on presence or absence. Researchers might also further investigate to what extent graphical ingredient representations reduce product neophobia.

From a methodological point of view, this research builds on previous packaging design studies by analyzing several variables simultaneously using a realistic depiction of the package (designed in collaboration with a food industry innovation project group and created by a computer graphic designer). Previous studies often use black-and-white drawings of objects [START_REF] Veryzer | The influence of unity and prototypicality on aesthetic responses to new product designs[END_REF] or unaltered photographs [START_REF] Hekkert | Most advanced, yet acceptable: typicality and novelty as joint predictors of aesthetic preference in industrial design[END_REF]. In this research, realistic designs allowed participants to evaluate an actual prototype as a potential entrant to the beverage market.

Implications for food marketers

Finally, from a managerial point of view, this research offers insight regarding the design of new food products and provides guidelines helpful in the selection of suitable flavors, labels, and packaging. Typicality, graphical representation, and packaging were influential factors in consumers' pleasantness and sensory perceptions. These factors also changed the perceived taste intensity the product. Graphic representation of sensory components influenced product evaluations and acceptability, which suggests new avenues for segmentation and positioning in the food industry. In practice, graphic representations may be particularly useful in conveying atypical taste combinations, which increases the acceptability of potential line extensions and encourages development of concepts directly linked to flavoring.

Limitations and future research

This research has several limitations that could open the door to further research. First, the studies did not account for several variables that could influence beliefs and evaluations of products, such as the brand name or typeface used on the label, as well as mediating variables such as perceived authenticity. The typography could be a particularly interesting variable to manipulate, in that certain fonts might be more appropriate for innovations, depending on the product category. The use of certain typographies likely creates greater congruence with an innovative connotation, as well as increases the significance of the social representation of the product for the consumer.

Secondly, Mueller, Lockshin, and Louviere (2009) recommend caution when using direct measures with visual packaging attributes, because their comparison to indirect discrete choice experiments finds a greater impact in discrete choice experiment, as well as significant preference heterogeneity. Further research therefore could therefore replicate the studies reported herein using this measurement method. In this sense, it seems relevant to discover if all the elements have effects on actual behaviors, such as purchases and price evaluations. Jaeger and Harker (2005) find that consumers are ready to pay twice as much for a kiwi with flesh, and two and half times more if the communication indicates the kiwi has not been genetically modified. Thus, product color, perceived newness, and informational stimuli all should have significant impacts on the price consumers are willing to pay.

Thirdly, further research should consider potential moderators, such as consumers' desire for food variety [START_REF] Van Trijp | Consumers' variety seeking tendency with respect to foods, measurement and managerial implications[END_REF] or the need to be unique [START_REF] Tian | Consumer's need for uniqueness: scale development and validation[END_REF]. Consumers with greater variety seeking tendencies are more susceptible to accept innovations, and the need to be unique might lead a consumer to buy innovative and surprising products [START_REF] Irmak | You like what I like, but I don't like what you like: uniqueness motivations in product preferences[END_REF].

Finally, it would be interesting to advance the analysis by combining this test of sensory evaluations based on extrinsic attributes of the product with a gustative evaluation that is based on the actual taste properties of the food product. Such studies could help delineate various perceptions and outline the reasons for confirmed or rejected expectations, which likely have correlative impacts on satisfaction.

H2:

  The presence (vs. absence) of a visual representation of product ingredients on product labels results in more positive expectations about (a) pleasantness, (b) taste, (c) color, and (d) smell. Study 1 examines the impact of flavor typicality (H1) and graphical representations of ingredients (H2) on consumers' sensory and pleasantness expectations. This research further investigates the influence of package typicality on consumer responses to a new product. The conceptual background is discussed next.

Table 1 Influence of ingredient type and presence on product evaluation

 1

	Independent Variables	Dependant Variables	Mean Level 1	Mean Level 2	F	P
		Pleasant to drink (PLeasantness)	3.04	2.63	18.773	0.000 ***
		Sickening (PL)	2.97	3.54	25.543	0.000 ***
	Simple effect	Fruity taste (TAste)	4.02	2.92	127.92	0.000 ***
		Disagreeable taste (TA)	2.70	3.04	11.069	0.000 ***
	Blackcurrant (1) / Caramel (2)	Original color (Color)	3.57	3.09	22.108	0.000 ***
		Distant color (CO)	3.84	3.16	45.435	0.000 ***
		Agreeable smell (SMell)	3.04	2.85	4.643	0.032 **
		Special smell (SM)	3.30	3.58	9.002	0.003 **
		Pleasant to drink (PLeasantness)	2.72	2.92	4.204	0.041 **
		Sickening (PL)	3.40	3.16	4.569	0.033 **
		Fruity taste (TAste)	3.41	3.51	1.092	0.297
	Simple effect	Disagreeable taste (TA)	2.96	2.81	2.122	0.146
	Absence (1) /	Original color (Color)	3.31	3.34	0.062	0.803
	Presence (2)	Distant color (CO)	3.57	3.43	1.705	0.192
		Agreeable smell (SMell)	2.90	2.98	0.672	0.413
		Special smell (SM)	3.38	3.50	1.671	0.197

TABLES


Amit Das

Amit Acharya

Pierre Suquet

Microstructure in plasticity without nonconvexity

come

Introduction

A long-standing goal of modern plasticity theory at the meso/macro scale is to model physically observed microstructure. In this paper, we define microstructure as spatial inhomogeneity in the plastic strain distribution and its development as the question of instability of a spatially uniform plastic strain distribution to small perturbations. In more common language this issue is also termed as the spontaneous emergence of patterns, and the models we develop and deal with here belong to the class of pattern-forming equations.

The minimum ingredients necessary to predict microstructure in plasticity are believed to be non-monotone equations of state arising from non-convex (incremental) energy [START_REF] Aifantis | On the microstructural origin of certain inelastic models[END_REF][START_REF] Ortiz | Nonconvex energy minimization and dislocation structures in ductile single crystals[END_REF], or reaction diffusion equations with Cahn-Hilliard (CH) or Ginzburg-Landau (GL) structure (Walgraef and Aifantis, 1985,I, II, III;[START_REF] Glazov | Self-organized dislocation structures (SODS) in fatigued metals[END_REF]. In this paper, we show that the simplest models derived from Phenomenological Mesoscale Field Dislocation Mechanics, i.e. PMFDM [START_REF] Acharya | Size effects and idealized dislocation microstructure at small scales: predictions of a phenomenological model of mesoscopic field dislocation mechanics: Part i[END_REF][START_REF] Acharya | New inroads in an old subject: plasticity, from around the atomic to the macroscopic scale[END_REF], building on work from the same group from 2001 onwards) produces spatial patterning with similarities to 'cell-wall' fatigue microstructures without any non-monotone equations of state but due to wave-propagative dislocation transport. An important motivation for our study is that (P)MFDM in its full-blown multidimensional system form, as well as very closely related, if not identical, models have been shown to produce the emergence of microstructural features without any non-convexity in extensive numerical computations [START_REF] Roy | Size effects and idealized dislocation microstructure at small scales: predictions of a phenomenological model of mesoscopic field dislocation mechanics: Part II[END_REF][START_REF] Limkumnerd | Mesoscale theory of grains and cells: crystal plasticity and coarsening[END_REF][START_REF] Chen | Bending crystals: Emergence of fractal dislocation structures[END_REF][START_REF] Choi | Is dislocation flow turbulent in deformed crystals?[END_REF][START_REF] Chen | Scaling theory of continuum dislocation dynamics in three dimensions: Self-organized fractal pattern formation[END_REF]. Another recent work with many mathematically similar features as PMFDM but more complicated constitutive structure related to multiple-slip behavior [START_REF] Xia | Computational modelling of mesoscale dislocation patterning and plastic deformation of single crystals[END_REF] shows the development of realistic microstructure related to plastic response. It is our intention here to try to understand this phenomenon in PMFDM in as simple a setting as possible.

We also demonstrate stick-slip behavior arising from a non-convex energetic constitutive statement for kinematic hardening, admittedly of undecidable physical origin. Combinations of dislocation transport with the resulting non-monotone back stress produce very rich behavior consisting of apparently random nucleation of fronts of plastic deformation and their motion under spatially uniform applied stress.

Model

Physical description

We adapt an ansatz producing an exact time-dependent, quasi-static, 1-dimensional problem of Field Dislocation Mechanics from [START_REF] Acharya | New inroads in an old subject: plasticity, from around the atomic to the macroscopic scale[END_REF]. Symbols with a subscript x represent a spatial partial derivative while a subscript t represents a temporal partial derivative. Consider a long cylinder of rectangular cross-section as shown in Fig. 1. The cylinder is subjected to simple shear by applying spatially uniform shear traction   T t on the top surface while holding the bottom surface fixed. It is assumed that at any time t all stress and distortion fields vary only along the axis of the cylinder. Consistent with the applied loading, it is assumed that the yz component of the total displacement gradient is the only non-zero one and it is homogeneous on the body. The only non-zero components of stress field are the shear stress   T t in the y direction on planes with normal in the z direction and its associated symmetry-related component. Likewise, the only non-zero component of plastic distortion is  which represents plastic shearing in the y direction on planes with normal in the z direction. The spatial gradient of plastic distortion x 

represents the only component of the (polar) dislocation density field representing screw dislocations with line and Burgers vector along the y direction. In this work we model plasticity at the mesoscale and consequently think of plastic strain rate produced by motion of the dislocation density field plus that due to the motion of unresolved 'statistical' dislocations, the latter modeled by more-or-less conventional constitutive assumptions from macroscopic plasticity theory. The governing equation for the evolution of plastic distortion  is derived here based on [START_REF] Acharya | New inroads in an old subject: plasticity, from around the atomic to the macroscopic scale[END_REF]. The dissipation, D , in this 1-d setting may be written as

  1 : 2 , 2 e t t B B D T dv dv               (1)
where T and e  are the yz component of the stress and elastic distortion respectively, and e    is the same component of the total displacement gradient tensor. The free energy density,  , is supposed to be of the form

    2 ˆ. 2 e x G          (2)
Here, The first term of (2), models the energy stored in the body due to elastic deformation. The second models the energy stored in dislocation cores, while the third term is associated with the phenomenological energy content of unresolved statistical density that produces kinematic hardening. Stress can be obtained from free energy density as follows:

 

.

e T      (3)
Substituting ( 2) and ( 3) in (1) we get

  . t x x t t B B B G D T dv dv dv              (4)
Using the governing equations of the mesoscale field dislocation mechanics model [START_REF] Acharya | New inroads in an old subject: plasticity, from around the atomic to the macroscopic scale[END_REF][START_REF] Acharya | Size effects and idealized dislocation microstructure at small scales: predictions of a phenomenological model of mesoscopic field dislocation mechanics: Part i[END_REF],it can be shown that in this simplified case the conservation law for Burgers vector content can be written as (see Appendix 1)

    , p x x t x p t x V L V L            (5)
where the source term, P L , arises in the governing equation as result of averaging [START_REF] Acharya | Size effects and idealized dislocation microstructure at small scales: predictions of a phenomenological model of mesoscopic field dislocation mechanics: Part i[END_REF], and represents the plastic strain rate produced by 'unresolved' statistical dislocation density. Substituting ( 5) in (4),

    , p p x x x x B B G D T V L d v V L d v                       (6) which implies     , p p x x x x B B G D T V L dv V L dv                       (7)
(where we have ignored a boundary term). Hence, the driving force for dislocation velocity, V , can be expressed as ,

x x x G V T                 (8)
and that for p L as

p G L T            . (9)
In ( 9), we retain the classical form of p L , where the driving force is dependent only on stress and back-stress, and ignore the contributions of core energy. This enables a direct comparison with the conventional plasticity model. Nevertheless, we have tested the dissipation (7) while using (12) and found that it remains nonnegative throughout the entire simulation for various model set-ups as described in Section 2.

Assuming a simple 'linear' kinetic relation for V based on (8):

1 ,

x x x G V T B                        ( 10
)
where the drag coefficient B  is proportional to the magnitude of the polar density x  ,the following expression for the evolution of plastic distortion tensor,  , is obtained from (5) :

P x t x x G T L B                  . ( 11
)
The material constant B has physical dimensions of stress time length  . Equation ( 11) is an averaged equation modeling plastic behavior at the meso-scale. We assume p L to be of 'standard' form

1 0 sgn m P G T G L T g                              0 0 , p s t x s y g g g L k g                        (12)
following conventional plasticity assumptions of models with a back stress and power-law kinetics. Here, 0

 is a reference strain rate and g is the strength of the material which evolves according to a gradient modified standard Voce-law [START_REF] Acharya | Grain-size effect in viscoplastic polycrystals at moderate strains[END_REF]. The saturation stress is s g , y  is the yield stress, and 0  is the Stage II hardening rate, 0 k is a parameter characterizing the hardening rate produced by the presence of polar density (containing a length scale), and m is the rate sensitivity.

An important goal for us in this paper is to understand what the addition of dislocation transport does to the conventional plasticity model in as simple a setting as possible. Thus, we would like to preserve the structure of (12). To simultaneously achieve this goal and ensure non-negative dissipation with our constitutive choices for V and p L would require that the specific free energy cannot be defined as a function on the current state but rather as a functional on histories through the following rate equation (cf. Rice, 1971, Sec. 2.4): ,,, .

    ˆ, , , , + ,
e e t t x x x x x x e p e x xx G V g G L g                                       (13)
We utilize a non-monotoneback-stress term, G    , in the governing equation whose form is sinusoidal with amplitude proportional to the wavelength. It is assumed to arise from a nonconvex energy contribution of the form

  sin 1 y G               ( 14
)
where  is a nondimensional parameter. Since  is small, the contribution of the   G  term in the energy is negligible. Thus, in practical terms, if back stress effects are determined from measurements of energy storage due to plastic strain (the part of it that is determined from the observable dislocation density field), then whether such terms may or may not be present in material behavior would be hard to decide. However, the back stress, G    , arising from such a small energy contribution can of course be of significant magnitude. An ODE of the form,

  , G f          ( 15
)
where f is a forcing function, is known to produce stick-slip behavior (e.g., [START_REF] Abeyaratne | Kinetics of materials with wiggly energies: theory and application to the evolution of twinning microstructures in a Cu-Al-Ni shape memory alloy[END_REF]. The particular form of the G    term does not have an influence as long as it is a high-frequency bounded oscillatory function with the same amplitude y  . Any high-frequency, bounded oscillatory function in the plastic strain with the same amplitude y  is expected to produce similar stick-slip behavior in the high frequency limit.

In recent experiments of micron-scale plasticity, stick-slip behavior has been observed [START_REF] Dimiduk | Scale-free intermittent flow in crystal plasticity[END_REF] and it has often been argued that its origins are stochastic [START_REF] Parthasarathy | Contribution to size effect of yield strength from the stochastics of dislocation source lengths in finite samples[END_REF]. We show here that such behavior can be obtained from purely deterministic considerations. However, the primary shortcoming of this model is that the physical origin of such a small amplitude augmentation of the energy is not clear at this point. Thankfully, such an augmentation of the theory is not essential for producing spatial microstructure as we show in this paper. In the following, we refer to such oscillatory kinematic hardening related energy and back stress by the term 'wiggly,' using terminology introduced by [START_REF] Abeyaratne | Kinetics of materials with wiggly energies: theory and application to the evolution of twinning microstructures in a Cu-Al-Ni shape memory alloy[END_REF] in a different context.

Non-dimensionalization of governing equations

We seek to non-dimensionalize the set of governing equations ( 11) and (12). Consider length, stress and time as the primary dimensions of the governing equations. The scaling parameter utilized for length variables is the length of the domain L , the stress variables is the yield stress y  , and the time variable is the reciprocal strain rate 1 0    . We define the following dimensionless variables using the scaling parameters: 0 0 , , , , .

p p y y y x T G L g x t t T G L g L                    (16) We have 0 . t t t t t t                     (17) Likewise, 1 , 1 . x x xx x x L L         (18)
Substituting( 16), ( 17) and ( 18) in the governing equation ( 11) we obtain

0 0 2 . x P y y x x t G T L BL L                              (19)
We rearrange ( 19) to get a non-dimensional form as follows:

* * , x P xx t G T L B                       ( 20
)
where * B and *

 are new constants defined as

* * 0 2 , . y y BL B L         (21) An important order-of-magnitude to keep in mind is that 2 y b L            (22)
so that a small value of   may be interpreted as working close to the macroscopic limit where the domain size is much larger than the interatomic distance.

The non-dimensional form of p L is given by 1 sgn .

m P G T G L T g                                    (23)
The non-dimensional form of the hardening rate equation is obtained from

0 0 0 0 s y p y x t s y g g k g L L g                                   ( 24
)
and then rearranging the terms to obtain

* * 0 0 * 1 p s x t s g g g L k g                         , ( 25
)
where * 0 k and * s g , and 0 q * are new constants defined as follows:

* * 0 0 0 0 , , . s s y y y k g k g L          (26)
For the sake of convenience we drop the tilde and asterisk subscripts in the nondimensional governing equations.

Notation

The following notation is followed in the numerical scheme described below.

  k xx h x  . The notations     k k h G x     and     2 2 k k h G x     represent partial

Numerical Scheme

We follow [START_REF] Acharya | On boundary conditions and plastic strain-gradient discontinuity in lower-order gradient plasticity[END_REF] in formulating the scheme utilized to deal with wave like response. This scheme has been subsequently utilized in Das et al. (2012) and [START_REF] Zhang | A single theory for some quasi-static, supersonic, atomic, and tectonic scale applications of dislocations[END_REF] for an extensive class of more complex problems in 1 and 2 space dimensions and time including nontrivial coupling with quasi-static and dynamic material deformation, some of which serve as verification and validation exercises for it, both individually and embedded in more complicated coupled algorithms. We have subsequently realized that the scheme is strongly related to a special and simple case of the robust methodology of [START_REF] Kurganov | Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton--Jacobi equations[END_REF] for hyperbolic conservation laws and Hamilton-Jacobi equations.

The essential idea of the scheme is to identify the first order wave-like, second-order diffusion-like and (ordinary differential equation) source-like responses in the problem from the linearization of the pde for  at each time level. Then at each spatial grid point an upwind direction is identified based on this exercise. In addition, time step constraints for wave, diffusion, and simple source behavior are evaluated at each spatial grid point based on standard stability considerations for consistent standard finite difference approximations for constant coefficient, linear equations describing such behaviors. Finally, the minimum time-step arising from these evaluations is adopted in the scheme.

The linearization of non-dimensionalized form (20), ignoring the source term, is performed formally by taking its first variation :

                          2 2 sgn . k k x h k k k k k t h h x x h x h k k x k k h h k x h k xx h x G x T x x x B x G x x B x x B                                                 (27)
The first term of ( 27) provides linearized wave-like behavior. A corresponding velocity

  k h c x is obtained as follows                               2 2 , sgn k k k t h h x h k k x h k k h h k x h k xx h k k x h k k k k h h x x h x c x x x G x x B x x B x G where c x T x x B                                                   (28)
To calculate  

    1 1 1 1 2 ( ) ( ) 2 ( ) 2 ( ) ( ) k k k h h x h k k k k h h h xx h x x x d x x x x d                  (29) Once   k h c x is obtained, depending on the sign of   k h c x the value of   k x h x  is updated
in accordance with the upwind scheme as follows:

            1 1 1 1 ( ) ( ) : 0 ( ) ( ) : 0 ( ) ( ) : 0 2 k k k k h h x h h k k k k h h x h h k k k k h h x h h x x x i f cx d x x x i f cx d x x x i f cx d                        (30)
where d is the element size.

The time step is computed as below, taking due account of the source term and linearized advection and diffusion:

            2 2 all 2 0.002 min , , , k k k k p h k k h x h h x h h d B dB t c x G x L x x x                              (31)
The following sequence of steps is followed to solve the governing equations( 20) and ( 25)

. The state at discrete time level k consists of nodal values of the discrete fields  and g

.                         1 1 1 2 1 1 1 ( ) 2 ( ) ( ) 0 0. 2 k k k k k k k k h x h h h k k k k h h h P k h h k k k P k k h h h h h G T x x x x t B x x x d L x if c x x x x L x t i f c x                                               (32)               1 1 0 0 . 1 k m k k h k k k h h s h k x h k k h s G T x g x g x g g x k x t g x g                                         (33) 1.5. Problem Set-up
The list of parameters selected for performing the simulations is shown in The initial condition specified for  is of the following form:

  Initial condition : , 0 sin 0 1, N x x L              (34)
where the chosen value of  and N is equal to 0.001 and 500, respectively. This reflects a very small amplitude with high wave-number, N , perturbation to the zero plastic strain initial condition. Our goal in all cases is to probe the stability of a spatially homogeneous initial profile to this particular type of perturbation. Neumann boundary conditions of 0

x   are applied at the ends of the domain. A linear ramp loading is specified for the simulations where the non-dimensional applied load parameter,   T t , increases uniformly from zero to a finite value of stress, unless otherwise mentioned.

The initial condition for g is assumed to be 1 (non-dimensional).

Since the model is rate-dependent, it is expected that the rate of loading will have an influence in the response of the governing equation. The effect of different rates of loading on the behavior of the governing equation is not pursued in this section because our primary interest has been to show the development of spatial microstructure in the model.

Clearly, the initial condition we choose has large polar dislocation content even though it derives from a negligible plastic distortion profile. From the point of view of probing stability of evolution of plastic deformation, this is, of course, perfectly acceptable. In all cases involving dislocation transport, the dislocation density profile rapidly subsides to be non-detectable to visual inspection on evolution from the initial condition, much before initial yield and subsequent inhomogeneity development. However, the seeds of the perturbation remain and without the perturbation in the initial condition no patterning develops. Thus we could as well have started with initial conditions as this slightlyevolved, almost-homogeneous plastic strain and dislocation density profile, if we knew their analytical form. Since we do not, we proceed as described in this paper.

In demonstrating our results, we often plot the spatial average of  and its maximum value

over

Results and discussion

Case I: Model with dislocation transport, wiggly back stress, and gradient hardening

The results of the simulations performed using the model described by ((20), ( 23),( 25)), i.e. 1 0 0 sgn 1

x P t x x m P p s t x s G T L B G T G L T g g g g L k g                                                             
are discussed in this section. T t reaches approximately a value of 2.0 before evolution occurs. This is due to the back-stress in the formulation -for 0   , 1 G    and the initial condition of 1 g  so that evolution occurs only when 2 T  for m small. This phenomenon is indicated in Fig. 2 (c) where the variation of average and maximum plastic distortion is plotted versus time. When the value of   T t increases beyond 2.0 the profile begins to rise and small kinks begin to develop in the profile as shown in Fig. 2 (a) (ii). These small kinks begin to grow as shown in Fig. 2 (a) (iii) and (iv) as the applied load increases, and develop into peaks representing inhomogeneous development in the  profile. All the peaks develop simultaneously at almost the same time. The pattern of the peaks appears to be repetitive along the domain. The profile continues to rise and the peaks smooth out as the value of   T t approximately reaches 2.07 . Once all the peaks smooth out the profile transforms into a nearly homogeneous form. The sequence of evolution depicted in Fig. 2 (a) (i) -(v) conforms to an almost vertical line in Fig. 2 (c) representing a rapid rise in plastic distortion for a minimal increase in applied stress. The profile retains its nearly homogenous form and rises uniformly during the remainder of the simulation as the applied stress increases to a value of 3.0 . This phenomenon is denoted by an inclined straight line in Fig. 2 (c). These plots demonstrate that this model is capable of producing spatial microstructure but stick-slip behavior is not predicted. As we shall see in Section 3, the simulations presented in Figure 2 (a-c) above for the parameter values mentioned do not produce converged results with respect to mesh refinement -the reliable conclusion that can be drawn from the specific simulation shown is the appearance of microstructures, but not the specific microstructure itself. However, this is simply a feature of the parameter regime we are operating in (close to the macroscopic limit) and not any specific ill-posedness associated with the model (or the numerical scheme) in more moderate circumstances corresponding to smaller domain sizes in comparison to the interatomic distance. A simulation is performed for a reduced wave number in the initial condition, N , equal to 5 and a higher value of *  , increased by a factor of 10 from that mentioned in Table 1. The results of the simulation, shown in Fig. 3, indicate the development of inhomogeneity and yield-like behavior. The magnitude of peaks developed in the  profiles is non uniform across the domain. Stick-slip phenomenon is not predicted. It will be shown later in Section 3 that this setup leads to a better convergence of results as compared to the model setup for larger domain size. , is illustrated from these results where a dormant phase (with no plastic growth) is followed by a phase of greater plastic growth (with significant movement and interaction of dislocations) and is again followed by a dormant phase. Overall, the completely deterministic evolution gives an impression of apparently stochastic behavior. The behavior predicted is reminiscent of the propagation of Luders' bands in metals. Given the strong discontinuities, both in space and time, that are predicted, it is highly unlikely to expect any type of convergence in these results with respect to mesh refinement. We present the results simply because of their 'remarkable' nature. that due to the motion of the unresolved statistical dislocations modeled by p L . In this section, we investigate a different model for the first part of the strain-rate, assuming that it does not result from transport, but rather is governed by the thermodynamic force    . A constitutive relation must be prescribed for this part of the strain-rate, replacing the kinetic relation (10) for V , and this is done by assuming a simple linear dependence on the thermodynamic force. The resulting set of equations used in the simulations is .

(v) 2.065 T  (vi) 3.00 T  (i) 0.2 T  (ii) 2.00 T  (iii) 2.03 T  (iv) 2.05 T  Fig. 2
(i) 0.3 T  (ii) 2.02 (iii) 2.03 T  (iv) 2.05 T  (v) 2.065 T  (vi) 3.00 T  Fig. 3
(i) 0.1 T  (ii) 2.0189 T  (iii) 2.0190 T  (iv) 2.02 T  (v) 2.365 T  (vi) 2.366 T  (vii) 2.694 T  (viii) 2.699 T  (ix) 2.857 T  (x) 2.862 T  Fig. 4
1 m t x x m s t x s G T G G T T B g G T g g g k g g                                                                             (36)
Note that these equations are apparently similar to ( 11)-( 12), with the noticeable difference that there is no transport term x  in the first term on the right-hand side of (36) 1 .

Figure 5(a) shows the evolution of the  profiles in time. Although a yield-like behavior similar to that in Fig. 2 (c) is observed the yielding happens comparatively earlier when the applied stress is approximately equal to 1.2 . This is because the extra plastic strain rate term (i.e. the first term on the right-hand side of (36) 1 ) is active without a yield threshold and even in the presence of homogeneous plastic strain. After yielding, the profile begins to move upwards and there is no development of peaks in the  profile. Hence, in the absence of dislocation transport in plastic strain rate, there is neither a development of large scale inhomogeneity nor a formation of ideal microstructure.  profiles that clearly demonstrate the dislocation microstructure that develops. Dense dipolar walls of positive and negative dislocations separated by spatially non-periodic 'cell-like' regions of relatively sparse dislocations density are observed. This microstructure bears some superficial similarity to the persistent slip-band microstructures that are observed in fatigue experiments (Mughrabi, Ackerman and Herz, 1979). A magnified view, shown in the section plots of Fig. 6 (c), suggests that The third microstructural length scale is the cell-size (regions of low dislocation density content). In all three cases there does not appear to be a fixed value for each length, but rather they vary in space and time. We think that these lengths are related to the nondimensional parameters *  , * B , and * 0 k , and suspect that there are asymptotic scaling laws to be uncovered (cf. [START_REF] Barenblatt | Scaling, self-similarity, and intermediate asymptotics: dimensional analysis and intermediate asymptotics[END_REF] in this context. It is possible to provide some mathematical understanding of the emergence of inhomogeneity in this case. Equation ( 37) is a degenerate, quasi-linear parabolic system that results in a nonlinear (not quasi-linear) first-order system of pde in g and x    for 0   , i.e. in the absence of core energy. The principal part of the linearization of this first- order system belongs to the class analyzed in Sec. 4.2.1 of [START_REF] Roy | Size effects and idealized dislocation microstructure at small scales: predictions of a phenomenological model of mesoscopic field dislocation mechanics: Part II[END_REF]. It was shown there that the problem is only weakly-hyperbolic (real eigenvalues, but not a full set of eigenvectors) when linearized about a homogeneous plastic strain profile, the case of interest here. As is known, weakly hyperbolic systems are unstable (Kreiss and Lorenz, 1999), and it is no surprise then that we see growth of perturbations. We note that the degeneracy of the parabolic regularization in our problem allows the principal part of the first-order problem to remain intact for linearization about the homogeneous plastic strain state. Thus, development of inhomogeneity is expected to be a generic feature of our model, unrelated to non-monotone equations of state. What the linear analysis does not say anything about is the long-time response of the growing perturbations, and our calculations here provide yet another confirmation of the conjecture in [START_REF] Roy | Size effects and idealized dislocation microstructure at small scales: predictions of a phenomenological model of mesoscopic field dislocation mechanics: Part II[END_REF] that because of the stabilizing feedback, through hardening due to the increase in  , the instability should be controlled. Perhaps more convincing is the fact that we have the degenerate parabolic regularization due to the core energy for 0   in (37) which provides particularly strong diffusive regularization in any region of sharp gradients in  (due to the multiplication by x  ), and therefore the  solutions of the model may be generically expected to remain smooth and  bounded. We remark that in problems where a system with more than one plastic strain component is involved, it is known that, at least in 1 spatial dimension and time, the system is weakly hyperbolic without core regularization even without any involvement from hardening [START_REF] Acharya | On an equation from the theory of field dislocation mechanics[END_REF], and again the development of microstructure is to be expected (cf. [START_REF] Chen | Scaling theory of continuum dislocation dynamics in three dimensions: Self-organized fractal pattern formation[END_REF].

  1 1 0 0 , 0, . 1 m x t x x m s t x s T T T B g g g T g k g g                                       (37)
2.5.Case V: Model with dislocation transport standard hardening with no non-monotone equations of state Fig. 7 (a) is a plot of the evolution of  when the wiggly back-stress term and 0 k is set to be equal to zero. The model is: The conventional plasticity model can be simulated using the following equations:

  1 1 0 , 0, . 1 m x t x x m s t s T T T B g g g T g g g                                     (
      1 1 0 ; 0 . 1 m t m s t s T t T t g T t g g g g g                                (39)
The result of the simulation performed using (39) is shown in Fig. 8 (a). The applied loading increases uniformly from 0 to 3 . The form of the initial  profile is retained during the entire simulation. A yield like behavior is observed where the profile remains stationary until the applied load reaches 0.95 and rises with further increase in loading.

There is no development of large-scale inhomogeneity in the  profile.

Limit of the models corresponding to macroscopic scales of observation

The constitutive model ( 11)-( 12) contains a small parameter  which measures the intensity of the core energy and, as explained in the discussion surrounding ( 22), its nondimensional counterpart   also measures the length scale of observation with respect to the interatomic distance. This core energy plays a (physical) regularizing role in that it helps to spread the plastic distortion  and avoids singularities in its gradient. However, when observed at large macroscopic scales, such smooth but large gradients appear as discontinuities. The objective of this section is to investigate, numerically, the behavior of the different models as the nondimensional parameter   goes to 0.

A careful inspection shows that there are at least 3 small parameters at play in the computed response of the different models. In addition to the parameter   , the other parameters are the wavelength of the initial condition (34), 1 N   , and the spatial discretization

1 h h d x x    .
We first discuss the effect of mesh refinement on the results obtained from the model governed by (( 20), ( 23),( 25)), i.e. Case I. A set of simulations is performed with three meshes of 10000, 20000 and 40000 elements for the initial condition prescribed by (34) for a wave number in the initial condition 5 N  and a value of   increased by a factor of 10 from that mentioned in Table 1. The results of these simulations are shown in Fig. 9 (a). The resulting profiles for different mesh sizes have similar form and it can be considered that convergence with the mesh size is attained at a rather coarse mesh size. Next we consider the macroscopic limit which corresponds to 1    as given in Table 1 and a high wave-number initial condition of 500 N 

. Simulations are again performed with the three different meshes. Evolution of the  profile for different mesh sizes is shown in Fig. 9 (b). A lack of convergence is observed, particularly between stress levels 2.0 and 2.1 when inhomogeneity develops in the profile. A comparatively higher number of peaks develop in the profile for the simulation performed with a finer mesh. Two conclusions can be drawn from these observations. First, for the parameters considered, a very fine mesh is required in order to obtain converged results with respect to the mesh size, if at all.

The second conclusion is that solutions become highly oscillatory when   becomes small, a situation reminiscent of that of homogenization [START_REF] Tartar | The General Theory of Homogenization[END_REF], or wiggly energies [START_REF] Abeyaratne | Kinetics of materials with wiggly energies: theory and application to the evolution of twinning microstructures in a Cu-Al-Ni shape memory alloy[END_REF]. Therefore a careful mathematical analysis of the weak limits of these solutions is required in order to derive the "effective" equations governing the evolution of the system in the limit as   goes to 0. This is beyond the scope of the present paper.

Next we perform the same test in the macroscopic limit for the model described in Case IV governed by (37) with   as specified in Table 1 and 500 N 

. The comparison of results obtained with different meshes in shown in Fig. 9(c). The absence of non-monotone equations of state in the model leads to good convergence even for a high wave number initial condition.

To quantify the differences in the results for different meshes the following error measures are computed. predicted by the conventional plasticity model.  Convergence is achieved for a low wave number initial condition and incorporation of an effective mesoscale core energy in the model. Convergence can be achieved for a high wave number initial condition in the absence of non-monotone equations of state.  In the limit as the regularization due to the core energy vanishes (or the macroscopic limit), the model with non-monotone equation of state shows oscillations and sensitivity to the mesh resolution and to the initial data which are typical of weak convergence of the solutions. The limit problem and the type of convergence for the solutions remain to be studied but cannot be obtained by simply setting 0   in the equations. By contrast, in the absence of non-monotone equation of state, convergence is achieved even for a high wave-number initial condition.

K K K K L K K L L K K K K L K K L L dx dx dx dx dx dx                                       (

Fig. 1

 1 Fig. 1 Cylinder subjected to simple shear

  derivatives of the function G with respect to  , evaluated at time level k at the spatial location. the value of p L and g at time level k and location h x .

  a time level k and at a spatial location h x the required values of central finite differences.

  Fig. 2 (a) is a plot of the evolution of  for the simulation performed with the prescribed initial condition (34). A linear ramp loading where the nondimensional applied load variable,   T t , increases uniformly from 0 to 3.0 is specified. The value of  chosen for this simulation is equal to 0.01. From the initial condition, the profile quickly transforms into an almost homogeneous profile shown in Fig. 2 (a) (i). The corresponding x  profile has a uniform value close to 0 as shown in Fig. 2 (b) (i). A yield- like behavior is observed, as expected from (23) where the  profile remains almost unaltered until  

Fig

  Fig. 2 (a) Evolution of  for the model with dislocation transport, wiggly back stress, and gradient hardening (Case I)

  Fig. 3 (a) Evolution of  for Case I with high   and low N

  Fig. 4 (a) Evolution of  for the model with dislocation transport, wiggly back stress, and standard hardening (Case II).

Fig. 4

 4 Fig. 4 (b) Variation of spatial average and maximum values of  with applied load (Case II)

  Fig. 5 (b) shows the variation of spatial average and maximum values of  with applied load. The wavy nature of the slope after yielding in these plots suggests the presence of stick-slip behavior.

Fig. 5

 5 Fig. 5 (a) Evolution of  for the model with wiggly back stress, gradient hardening but no dislocation transport (Case III)

  Development of inhomogeneous profiles is observed. Fig. 6 (d) shows the variation of spatial average and maximum values of  with applied load. No stick-slip behavior is observed. Figs. 6 (b) (ii) -(vi) show the x

Fig. 5

 5 Fig. 5 (b) Variation of spatial average and maximum values of  with applied load (Case III)

Fig. 6

 6 Fig. 6 (a) Evolution of  for the model with dislocation transport and gradient hardening without non-monotone equations of state (Case IV)

Fig. 6

 6 Fig. 6 (d) Variation of spatial average and maximum values of  with applied load (Case IV)

  Fig. 7 (a) Evolution of  for the model with dislocation transport, standard hardening and no non-monotone equations of state (Case V)

  Fig. 7 (b) Variation of spatial average and maximum values of  with applied load (Case V)

Fig. 8

 8 Fig. 8 (c) Variation of spatial average and maximum values of  with applied load (Case VI)

FigFig

  Fig. 9 (a) Convergence of  evolution for Case I with high value of *  and low N

Table 1 .

 1

	Parameter	Description	Value
	m 	rate sensitivity non-convex energy parameter	0.03 0.01
	* * B 	scaled drag coefficient scaled dislocation core energy parameter	1   5 3.28 10 
	k	0	*	scaled hardening parameter	400
	g	s	*	scaled saturation stress	3.22
		0	*	hardening parameter scaled stage II	7.84

Table 1 :

 1 List of non-dimensional model parameters Lengths are measured in units of L , time in units of 1

	0    , and stress in terms of y  .

  the domain, denoted by avg

	 and max  respectively, as functions of time or applied
	load. avg  is computed as follows:
		k avg		1  P h 	k P 	  h x	,			(35)
	where P is the total number of nodes and avg  refers to k avg  viewed as a function of k (or
	discrete time or load). max k  represents the maximum value of ,		max k		max h		k	  h x	, over
	the entire set of spatial nodes at the th k time level, and a similar interpretation as before
	applies for max

 . Step-like profiles in avg  beyond initial yield reflect stick-slip behavior on the average in the domain. (Near) discontinuous behavior in max  different from the avg  variation reflects spatially localized stick-slip behavior.

Acknowledgment

AA acknowledges support in part from grants NSF-CMMI-1435624, NSF-DMS-1434734, and ARO W911NF-15-1-0239. PS acknowledges the support of the Labex MEC and of A*Midex through grants ANR-11-LABX-0092 and ANR-11-IDEX-0001-02.

Appendix1

Simplified one-dimensional form of conservation law of Burgers vector content

The conservation law for Burgers vector content in mesoscale field dislocation mechanics is given by

where  is the mesoscale Nye tensor, V is the mesoscale averaged dislocation velocity vector, and p L is the plastic strain rate produced by unresolved, at the mesoscale, and hence 'statistical' dislocations [START_REF] Acharya | Size effects and idealized dislocation microstructure at small scales: predictions of a phenomenological model of mesoscopic field dislocation mechanics: Part i[END_REF]